On the Bahri–Lions conjecture for elliptic equations with non-symmetric nonlinearities

Abstract

We prove the existence of infinitely many solutions for a class of elliptic Dirichlet problems with non-symmetric nonlinearities. In particular, in suitable domains of ${\mathbb {R}}^n$ with $n\ge 3$, this result gives a positive answer to a well known conjecture formulated by A. Bahri and P.L. Lions. The proof is based on a minimization method which does not require the use of techniques of deformation from the symmetry. This method allows us to piece together solutions of Dirichlet problems in suitable subdomains, so we obtain infinitely many nodal solutions with a prescribed nodal structure.

1 Introduction

In this paper we are concerned with Dirichlet problems of the form

$$\begin{aligned} -\Delta u=|u|^{p-1}u+\psi \quad \text{ in } \Omega ,\qquad u=0\quad \text{ on } \partial \Omega \end{aligned}$$

(1.1)

where $\Omega$ is a bounded domain of ${\mathbb {R}}^n$ with $n\ge 1$, $\psi \in L^2(\Omega )$, $p>1$ and $p<\frac{n+2}{n-2}$ when $n\ge 3$.

The solutions of problem (1.1) are the critical points of the energy functional $E_\psi :H^1_0(\Omega )\rightarrow {\mathbb {R}}$, defined by

$$\begin{aligned} E_\psi (u)=\frac{1}{2} \int _\Omega |{\nabla }u|^2dx-\frac{1}{p+1}\int _\Omega |u|^{p+1}dx-\int _\Omega \psi u\, dx \end{aligned}$$

(1.2)

where, under our assumptions, the exponent $p+1$ is less than the critical Sobolev exponent $2^*=\frac{2n}{n-2}$ for $n\ge 3$.

If $\psi \equiv 0$ in $\Omega$, the functional $E_\psi$ is even, so the equivariant Lusternik–Schnirelmann theory for ${\mathbb {Z}}_2$-symmetric sets may be applied and guarantees the existence of infinitely many solutions (see for instance [1, 3, 9, 18,19,20, 29, 30, 37, 39]).

A natural question, which goes back to the beginning of the eighties, is whether the infinite number of solutions still persists for $\psi \not \equiv 0$.

In particular, this question was raised to the attention by Rabinowitz in his monograph on minmax methods (see [39, Remark 10.58]). In [4] Bahri proved that, if $n\ge 3$ and $1<p<\frac{n+2}{n-2}$, then there exists an open dense set of $\psi$ in $L^2(\Omega )$ such that problem (1.1) admits infinitely many solutions. In [8] Bahri and Lions proved that, if $n\ge 3$ and $1<p<\frac{n}{n-2}$, then problem (1.1) admits infinitely many solutions for every $\psi \in L^2(\Omega )$. These results suggest the following conjecture, proposed by Bahri and Lions in [8]: the multiplicity result obtained in [8] holds also under the more general assumption $1<p<\frac{n+2}{n-2}$.

In the present paper we prove that, if the domain $\Omega$ is a cube of ${\mathbb {R}}^n$, then problem (1.1) has infinitely many solutions for every $\psi \in L^2(\Omega )$. Thus, for $n\ge 3$, our result shows that the Bahri–Lions conjecture is true at least when $\Omega$ is a cube of ${\mathbb {R}}^n$.

In order to show that the infinite number of solutions we have for $\psi \equiv 0$ persists under perturbations, a detailed analysis was originally carried on in [2, 3, 5,6,7,8, 26, 31, 32, 38, 41, 45] by Ambrosetti, Bahri, Berestycki, Ekeland, Ghoussoub, Krasnoselskyii, Lions, Marino, Prodi, Rabinowitz, Struwe and Tanaka by introducing new perturbation methods.

More recently, a new approach to tackle the break of symmetry in elliptic problems has been developed by Bolle, Chambers, Ghoussoub and Tehrani (see [10, 11, 17]). However, that approach (which works also for more general nonlinear problems) did not allow to solve the Bahri–Lions conjecture.

Related results can be found also in other, more recent, papers (see for example [40] and references therein).

In the present paper we develop a method introduced in [34] in order to construct infinitely many nodal solutions of problem (1.1), having a prescribed nodal structure.

The idea is to piece together the solutions of Dirichlet problems in suitable subdomains of $\Omega$. A similar idea has been first used by Struwe in earlier papers (see [41,42,43] and references therein). We consider as nodal regions some subdomains of $\Omega$ that are deformations of cubes by suitable Lipschitz maps (so we obtain nodal solutions having a “check” nodal structure). Notice that Lipschitz conditions combined with the covering of ${\mathbb {R}}^n$ by cubes with vertices in ${\mathbb {Z}}^n$ have been also used in some recent papers by Rabinowitz and Byeon in order to construct solutions with a prescribed pattern for the Allen-Cahn model equation (see [14, 15] and references therein).

The main result of the present paper is stated in Theorem 2.1 (which is a direct consequence of Proposition 2.2) and says that if $\Omega$ is a cube of ${\mathbb {R}}^n$, $n\ge 1$, $p>1$ and $p<\frac{n+2}{n-2}$ when $n>2$, then for all $\psi \in L^2(\Omega )$ there exist infinitely many nodal solutions of problem (1.1), having as nodal structure suitable partitions of $\Omega$ in subdomains that are Lipschitz deformations of arbitrarily small cubes. More precisely, in Proposition 2.2 we prove that there exists ${\bar{k}}\in {\mathbb {N}}$ such that for all positive integer $k\ge {\bar{k}}$ there exist at least two solutions $u_k(x)$ and $v_k(x)$ of problem (1.1) such that the nodal regions of the functions $u_k\left( \frac{x}{k}\right)$ and $v_k\left( \frac{x}{k}\right)$, after translations, tend to the cube $\Omega$ as $k\rightarrow \infty$. Moreover, the number of nodal regions of $u_k$, $v_k$ and their energy $E_\psi (u_k)$, $E_\psi (v_k)$ tend to infinity as $k\rightarrow \infty$, while the size of the nodal regions tends to zero.

Notice that, in dimension $n=1$, the existence of infinitely many solutions for all $\psi$ in $L^2(\Omega )$ follows from a result obtained by Ehrmann in [25] (see also [24, 28] for related results). However, the method used by Ehrmann relies on a shooting argument, typical of ordinary differential equations, combined with counting the oscillations of the solutions in the interval $\Omega$. On the contrary, in the present paper we use a method which is more similar to the one introduced by Nehari in [35], that can be in a natural way extended to the case $n>1$. In fact, Nehari’s method was used by Coffman in [19, 20] and, independently, by Hempel in [29, 30] to study an analogous problem for partial differential equations.

More recently, Nehari’s method has been used also by Conti, Terracini and Verzini to study optimal partition problems, existence of changing sign solutions etc. (see [21,22,23, 46]).

Let us remark that Nehari consider an odd differential operator (so the corresponding energy functional is even) and prove that for every positive integer k there exists a solution having exactly k nodal regions. On the contrary, as Ehrmann in [25], we find solutions with a large number of nodal regions. However, let us point out that our multiplicity result is sharp because, as we proved in [34, Proposition 3.5], $\psi$ can be chosen in $L^2(\Omega )$ in such a way that problem (1.1) does not have solutions with a small number of nodal regions: more precisely, for every positive integer h there exists $\psi _h$ in $L^2(\Omega )$ such that every solution of problem (1.1) with $\psi =\psi _h$ has at least h nodal regions.

Now, let us describe the method we use to prove our result. For every cube $\Omega$ of ${\mathbb {R}}^n$ and every positive integer k, let us consider the $k^n$ cubic open subdomains $C_1^k,C_2^k,\ldots ,C_{k^n}^k$, having all the same size, such that $\overline{\Omega }=\cup _{i=1}^{k^n}{\overline{C}}_i^k$. So, these subdomains are pairwise disjoint and, for all $i\in \{1,\ldots ,k^n\}$ the cube $kC_i^k$ is a translation of the cube $\Omega$.

Moreover, for all $L\in ]0,1[$, let us consider the set $D_L$ of all the deformations $T:\overline{\Omega }\rightarrow \overline{\Omega }$ such that T differs from the identity map in $\overline{\Omega }$ by a Lipschitz function with Lipschitz constant L, $T(\overline{\Omega })=\overline{\Omega }$ and $T(F)=F$ for every face F of the cube $\overline{\Omega }$. Notice that, since $L\in ]0,1[$, every deformation $T\in D_L$ is a bilipschitz map in $\overline{\Omega }$. Then, for all T in $D_L$ and k in ${\mathbb {N}}$, by using a Nehari type minmax argument in every subdomain $T(C_i^k)$ with $i\in \{1,\ldots ,k^n\}$, we construct two distinct nodal functions $u^T_k$ and $v^T_k$ in $H^1_0(\Omega )$ whose nodal regions are the subdomains $T(C_i^k)$, for $i=1,\ldots ,k^n$, and such that, for k large enough, $u^T_k$ and $v^T_k$ satisfy equation (1.1) in each nodal region and are solutions of the Dirichlet problem (1.1) in $\Omega$ when, in addition, they satisfy a suitable stationary property. Moreover, the construction of $u_k^T$ and $v_k^T$ shows that $v^T_k$ behaves as $-u^T_k$ when $k\rightarrow \infty$.

Now, for all $k\in {\mathbb {N}}$, we minimize the energy functional $E_\psi$ in the set $\{u^T_k\,\ T\in D_L\}$; moreover, we show that, if the minimum is achieved by a map $T^L_k$ in $D_{L_k}$ with $L_k\in ]0,L[$, then the corresponding function $u^{T_k^L}_k$ satisfies the stationarity condition which allows us to conclude that it is a solution of problem (1.1) for k large enough.

Indeed, we show that there exists a sequence $(L_k)_k$ of positive numbers such that $\lim _{k\rightarrow \infty }L_k=0$ and $T^L_k\in D_{L_k}$ $\forall k\in {\mathbb {N}}$, so $L_k\in ]0,L[$ for k large enough and the solution $u_k=u^{T_k^L}_k$ satisfies all the assertions of Proposition 2.2 (in analogous way one can construct the solutions $v_k$ that behaves as $-u_k$ when $k\rightarrow \infty$).

In particular, we obtain that $T^L_k$ tends as $k\rightarrow \infty$ to the identity map in $\overline{\Omega }$ and that the rescaled nodal regions $kT^L_k(C_i^k)$, after translations, tend to the cube $\Omega$ as $k\rightarrow \infty$, uniformly with respect to $i\in \{1,\ldots ,k^n\}$.

The existence of such a sequence $(L_k)_k$, which plays a crucial role in the proof, is strictly related to a minimality property of the cubes in ${\mathbb {R}}^n$. In fact, the functions $u^T_k\left( \frac{x}{k}\right)$, suitably rescaled, tend as $k\rightarrow \infty$ to solutions of the equation (1.1) with $\psi =0$. Therefore, since the effect of the term $\psi$ tends to vanish as $k\rightarrow \infty$, the rescaled nodal regions $kT^L_k(C^k_i)$, suitably translated, tend to polyhedra as $k\rightarrow \infty$. Among these polyhedra, the cubes of ${\mathbb {R}}^n$ are the unique minimizers of the “shape factor” ${\varphi }(\chi )$ defined by

$$\begin{aligned} {\varphi }(\chi )=m(\chi )|\chi |^{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) }, \end{aligned}$$

(1.3)

where $|\chi |$ is the volume of $\chi$ and

$$\begin{aligned} m(\chi )=\min \left\{ \int _\chi |{\nabla }U |^2dx\,\ U\in H^1_0(\chi ),\ \int _\chi |U|^{p+1}dx=1\right\} \end{aligned}$$

(1.4)

(notice that ${\varphi }(\chi )$ depends only on the shape of $\chi$ and not on its size because it is invariant with respect to translations and rescaling of $\chi$). Therefore, taking into account the asymptotic behaviour of $E_\psi (u^{T^L_k}_k)$ as $k\rightarrow \infty$, the minimality of $T^L_k$ implies that ${\varphi }(k T^L_k(C^k_i))\rightarrow \overline{{\varphi }}$, where $\overline{{\varphi }}$ denotes the shape factor of every cube of ${\mathbb {R}}^n$, while the volumes $|kT^L_k(C^k_i)|\rightarrow 1$ as $k\rightarrow \infty$, uniformly with respect to $i\in \{1,\ldots ,k^n\}$.

As a consequence, taking also into account the conditions of $T^L_k$ on $\partial \Omega$, we infer that, after translations, the rescaled nodal regions $kT^L_k(C^k_i)$ tend to $\Omega$ as $k\rightarrow \infty$ and there exists a sequence $(L_k)_k$ having the desired properties.

It is clear that our method does not require techniques of deformations from the symmetry and may be applied to more general problems. For example, it may be easily adapted to deal with the case where in problem (1.1) the nonlinear term $|u|^{p-1}u$ is replaced by $c_+(u^+)^p-c_-(u^-)^p$ with $c_+$ and $c_-$ two positive constants. Moreover, this method may be adapted to work even in case of nonlinear elliptic equations involving critical Sobolev exponents. For example, it allows us to obtain in this case a multiplicity result similar to Theorem 2.1, which is announced in Theorem 3.18.

Now, we can compare results and methods of the present paper with those of [34] in order to state clearly a relation between these papers.

In both papers the starting point of our approach is to observe that, for a given $f\in L^2(\Omega )$, there exists a mountain pass solution in every sufficiently small subdomain of $\Omega$. Then, we decompose the domain $\Omega$ into disjoint union of small subdomains which are given by the images of disjoint union of small cubes through a class of admissible Lipschitz maps. Next, on each subdomains we dispose mountain pass solutions, which are perturbations of (positive or negative) mountain pass solutions of problem (1.1) for $f=0$ (in such a way that the solutions in adjacent subdomains have different signs). Finally, we minimize the energy functional among the class of admissible Lipschitz maps and show that, if a minimizing map is (in a suitable sense) in the interior of the class of the admissible maps, the normal derivatives of the corresponding functions on the interfaces of the subdomains are equal and so we obtain infinitely many solutions (because the cubes may be arbitrarily small).

However, even if the scheme of our approach is similar in both papers, the class of Lipschitz maps considered in [34] is quite different from the class $D_L$ we use in the present paper. In fact, in [34] we denote by $P_k$ the union of all the cubes with sides of length $\frac{1}{k}$ and vertices in $\frac{1}{k}{\mathbb {Z}}^n$, that are enclosed in $\Omega$ and, for $L'>1$, we consider the class ${{\mathcal {C}}}_{L'}(P_k,\overline{\Omega })$ consisting of the bilipschitz maps between $P_k$ and $\overline{\Omega }$ with Lipschitz constants in $\left[ \frac{1}{L'},L'\right]$. Moreover, we show that, if the minimal energy is achieved by a bilipschitz map $T^{L'}_k$ with Lipschitz constants in $\left]\frac{1}{L'},L'\right[$, the corresponding function $u_k^{T_k^{L'}}$ is a solution of problem (1.1).

Thus, in [34] we prove the existence of infinitely many solutions under a geometric condition on the domain $\Omega$, which guarantees that, for a suitable choice of $L'>1$, the minimum is achieved by a map $T^{L'}_k$ which, for k large enough, belongs to ${{\mathcal {C}}}_{L''}(P_k,\overline{\Omega })$ for some $L''\in \left]\frac{1}{L'},L'\right[$.

This geometric condition may be easily checked in dimension $n=1$ because, since for all $L'>1$ the minimality of $T_k^{L'}$ implies that the subdomains tend to have all the same size as $k\rightarrow \infty$, for $n = 1$ we infer that $T_k^{L'}\in {{\mathcal {C}}}_{L'_k}(P_k,\overline{\Omega })$, where $L'_k\in [1,L']$ $\forall k\in {\mathbb {N}}$ and $L'_k\rightarrow 1$ as $k\rightarrow \infty$.

On the contrary, for $n >1$ it is difficult to check this geometric condition because, even if the subdomains tend to have all the same size, for $n>1$ they might have a shape very different from the cubes, so it is difficult to control the Lipschitz constants of the map $T^{L'}_k$ as $k\rightarrow \infty$.

For this reason in the present paper we use the class $D_L$ of admissible Lipschitz maps.

Moreover, let us point out that the approach used in the present paper and the class $D_L$ of Lipschitz maps work not only for cubes but also for other domains that can be decomposed into disjoint union of translations of scaling domains (for example, equilateral triangles, triangular prisms, rectangles, rectangular parallelepipeds, etc.).

In fact, in order to apply our method it is sufficient that the domain $\Omega$ admits tessellations by arbitrarily small congruent subdomains, symmetric by reflection with respect to all the interfaces and colourable with two colors in such a way that adjacent subdomains have different colors, so that we can have a criterion to choose the sign in every subdomain and the functions have the same normal derivatives on their interfaces (for example, tessellations by hexagons or hexagonal prisms do not work in our arguments).

In the present paper we describe our method only for cubic domains, for the sake of simplicity, because for the cubes the same arguments hold in every dimension n (on the contrary, for example, the tetrahedra do not work for $n=3$ as the equilateral triangles do for $n = 2$).

2 Variational framework and statement of the main results

Our aim is to prove the following theorem.

Theorem 2.1

Let $\Omega$ be a cube of ${\mathbb {R}}^n$ with $n\ge 1$, let $p>1$ and $p<\frac{n+2}{n-2}$ when $n\ge 3$. Then, for every $\psi \in L^2(\Omega )$, problem (1.1) admits infinitely many solutions.

Without any loss of generality, we can assume that

$$\begin{aligned} \Omega =\{x=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n\,\ 0<x_i<1\ \text{ for } i=1,\ldots ,n\}. \end{aligned}$$

(2.1)

For all positive integer k and for all $z\in {\mathbb {Z}}^n$, let us set

$$\begin{aligned} C_z^k=\frac{1}{k} (z+\Omega )\ \text{ and } \ \sigma (z)=(-1)^{\sum _{i=1}^nz_i} \end{aligned}$$

(2.2)

(thus, in particular, we have $C_0^1=\Omega$).

Notice that for all $k\in {\mathbb {N}}$ we have $C_z^k\subseteq \Omega$ if and only if $0\le z_i\le k-1$ for $i=1,\ldots ,n$; moreover, if we set

$$\begin{aligned} Z_k=\{z=(z_1,\ldots ,z_n)\in {\mathbb {Z}}^n\,\ 0\le z_i\le k-1\ \text{ for } i=1,\ldots ,n\}, \end{aligned}$$

(2.3)

we have

$$\begin{aligned} \overline{\Omega }=\bigcup _{z\in Z_k}{\overline{C}}_z^k\ \text{ and } \ C_z^k\cap C_{z'}^k=\emptyset \ \text{ for } \ z\ne z'\ (z,z'\in Z_k). \end{aligned}$$

(2.4)

Then, the following proposition holds (it obviously implies Theorem 2.1).

Proposition 2.2

Under the assumptions of Theorem 2.1, if $\Omega$ is the cube (2.1), for all $\psi \in L^2(\Omega )$ there exists ${\bar{k}}\in {\mathbb {N}}$ such that for every $k\ge {\bar{k}}$ problem (1.1) admits two solutions $u_k$ and $v_k$ having the following properties (here we consider $u_k$ and $v_k$ extended by the value zero in ${\mathbb {R}}^n\setminus \Omega$). For all $k\ge {\bar{k}}$ there exist two bilipschitz maps $T_{k,u},T_{k,v}:\Omega \rightarrow \Omega$ (with Lipschitz constants independent of k) such that for every choice of $z^k$ in $Z_k$ the functions $U_{z^k}$ and $V_{z^k}$ defined by

$$\begin{aligned}{} & {} U_{z^k}(x)=\frac{\sigma (z^k)}{k^{\frac{2}{p-1}}}u_k\left[ \frac{x}{k}+T_{k,u}\left( \frac{z^k}{k}\right) \right] \qquad \forall x\in {\mathbb {R}}^n,\ \forall k\ge {\bar{k}}, \end{aligned}$$

(2.5)

$$\begin{aligned}{} & {} V_{z^k}(x)=\frac{-\sigma (z^k)}{k^{\frac{2}{p-1}}}v_k\left[ \frac{x}{k}+T_{k,v}\left( \frac{z^k}{k}\right) \right] \qquad \forall x\in {\mathbb {R}}^n,\ \forall k\ge {\bar{k}}, \end{aligned}$$

(2.6)

restricted to $\Omega$, both converge as $k\rightarrow \infty$ to a positive solution of problem (1.1) with $\psi \equiv 0$ in $\Omega$, satisfying

$$\begin{aligned} E_0(U)=\min \{E_0(U)\,\ U\in H^1_0(\Omega )\setminus \{0\},\ E'_0(U)[U]=0\}. \end{aligned}$$

(2.7)

Moreover, the sequences $(T_{k,u})_k$ and $(T_{k,v})_k$ both converge to the identity map uniformly in $\Omega$, while the domains $k\left[ T_{k,u}\left( C^k_{z^k}\right) - T_{k,u}\left( \frac{z^k}{k}\right) \right]$ and $k\left[ T_{k,v}\left( C^k_{z^k}\right) - T_{k,v}\left( \frac{z^k}{k}\right) \right]$ tend to $\Omega$ as $k\rightarrow \infty$ for every choice of $z^k$ in $Z_k$.

The proof is reported in Sect. 3.

In order to prove Theorem 2.1 and Proposition 2.2, we proceed as follows. For every $t\in [0,1]$ and $i\in \{1,\ldots ,n\}$, let us consider the set

$$\begin{aligned} F_i^t=\{(x_1,\ldots ,x_n\}\in \Omega \,\ x_i=t\} \end{aligned}$$

(2.8)

(in particular, if $t=0$ or $t=1$, $F_i^t$ is a face of the cube $\Omega$).

Now, let us fix $L\in ]0,1[$ and consider the set $D_L$ of the admissible deformations of $\overline{\Omega }$ defined by

$$\begin{aligned} \begin{aligned} D_L=\{ T:\overline{\Omega }\rightarrow \overline{\Omega }\,\ {}&T( \overline{\Omega })=\overline{\Omega },\ T(F_i^t)=F^t_i\ \text{ for } t=0,1,\ i=1,\ldots ,n,\\&|T(x)-T(y) -x+y|\le L|x-y|\ \forall x,y\in \overline{\Omega }\}. \end{aligned} \end{aligned}$$

(2.9)

Notice that for every deformation $T\in D_L$ one can write $T(x)=I(x)+S(x)$ where $I(x)=x$ $\forall x\in \overline{\Omega }$ and $S:\overline{\Omega }\rightarrow {\mathbb {R}}^n$ is a Lipschitz continuous function with Lipschitz constant L. Moreover, we have

$$\begin{aligned} (1-L)|x-y|\le |T(x)-T(y)|\le (1+L)|x-y|\qquad \forall x,y\in \overline{\Omega }\end{aligned}$$

(2.10)

where $1-L>0$ because we assumed $L\in ]0,1[$. Thus, T is invertible and both T and $T^{-1}$ are Lipschitz continuous functions in $\overline{\Omega }$.

Other important consequences of the definition of $D_L$ are presented in next proposition where we describe some geometrical properties of the deformations $T(F^t_i)$ of the sets $F^t_i$ with respect to the straight lines orthogonal to $F^t_i$ (these properties motivate the introduction of this class of admissible deformations).

Proposition 2.3

Let $T\in D_L$ and $L\in ]0,1[$. Then

(a):

for all $t\in [0,1]$, $i\in \{1,\ldots ,n\}$ and $y\in \Omega$ there exists a unique $x\in F^t_i$ such that $P_i\circ T(x)=P_i(y)$, where $P_i$ denotes the orthogonal projection of ${\mathbb {R}}^n$ on the subspace $\{x=(x_1,\ldots ,x_n)\in {\mathbb {R}}^n\,\ x_i=0\}$ (that is, every straight line orthogonal to $F^t_i$ meets $T(F^t_i)$ in a unique point);

(b):

for all $t'$, $t''$ in [0, 1] such that $t'<t''$ and for all $x'\in F^{t'}_i$ and $x''\in F^{t''}_i$ such that $P_i\circ T(x')=P_i\circ T(x'')$, we have $T_i(x')<T_i(x'')$ (that is, the deformation $T(F^t_i)$ of the set $F^t_i$ meets every straight line orthogonal to $F^t_i$ in a unique point whose $i^{\textrm{th}}$ coordinate increases as t increases).

Proof

In order to prove (a), first notice that, for all $t\in [0,1]$, $i\in \{1,\ldots ,n\}$ and $y\in \Omega$, there exists $x\in F^t_i$ such that $P_i\circ T(x)=P_i(y)$.

In fact, let us consider the function $P_i\circ T:F^{t}_i\rightarrow F^0_i$, which is a continuous function satisfying

$$\begin{aligned} P_i\circ T(F^{t} _i\cap F^0_j)\subseteq F^0_i\cap F^0_j,\quad P_i\circ T(F^{t}_i\cap F^1_j)\subseteq F^0_i\cap F^1_j \qquad \forall j\in \{1,\ldots ,n\}\setminus \{i\}.\nonumber \\ \end{aligned}$$

(2.11)

Therefore, since $P_i(y)\in F^0_i$, there exists $x\in F^t_i$ such that $P_i\circ T(x)= P_i(y)$ (as follows from [33]).

Now, let us prove that such a x is unique. Arguing by contradiction, assume that there exists another ${\tilde{x}}$ in $F^{t}_i$, ${\tilde{x}}\ne x$, such that $P_i\circ T({\tilde{x}})=P_i(y)$, which implies

$$\begin{aligned}{}[T(x)-T({\tilde{x}})]\cdot (x-{\tilde{x}})=0. \end{aligned}$$

(2.12)

Since $T\in D_L$ with $L\in ]0,1[$, we infer that

$$\begin{aligned} |T(x)-T({\tilde{x}})+{\tilde{x}} -x|\le L|x-{\tilde{x}}| \end{aligned}$$

(2.13)

and, as a consequence,

$$\begin{aligned} |T(x)- T({\tilde{x}})+{\tilde{x}}-x|^2<|x-{\tilde{x}}|^2 \end{aligned}$$

(2.14)

because $x\ne {\tilde{x}}$. On the other hand, from (2.12) we obtain

$$\begin{aligned} \begin{aligned} |T(x)- T({\tilde{x}})+{\tilde{x}}-x|^2&= |T(x)- T({\tilde{x}})|^2+|{\tilde{x}}-x|^2 -2 [T(x)-T({\tilde{x}})]\cdot (x-{\tilde{x}})\\&= |T(x)- T({\tilde{x}})|^2+|{\tilde{x}}-x|^2\\&> |{\tilde{x}}-x|^2 \end{aligned} \end{aligned}$$

(2.15)

in contradiction with (2.14).

Thus, (a) is completely proved.

In order to prove (b), we argue again by contradiction and assume that there exist $t',t''$ in [0, 1] such that $t'<t''$ and $x'\in F^{t'}_i$, $x''\in F^{t''}_i$ such that

$$\begin{aligned} P_i\circ T(x')=P_i\circ T(x'')\ \text{ and } \ T_i(x')\ge T_i(x''). \end{aligned}$$

(2.16)

Notice that (2.16) implies

$$\begin{aligned}{}[T(x')-T(x'')]\cdot (x'-x'')\le 0. \end{aligned}$$

(2.17)

Therefore, we obtain

$$\begin{aligned} \begin{aligned} |T(x')- T(x'')+ x''-x'|^2&= |T(x')- T( x'')|^2+|x'-x''|^2 -2 [T(x')-T(x'')]\cdot (x'-x'')\\&\ge |T(x')- T(x'')|^2+|x'-x''|^2\\&\ge |x'-x''|^2. \end{aligned} \end{aligned}$$

(2.18)

On the other hand, since $T\in D_L$ with $L\in ]0,1[$ and $x'\ne x''$, we infer that

$$\begin{aligned} T(x')- T(x'')+ |x''-x'|^2\le L^2|x'-x''|^2<|x'-x''|^2 \end{aligned}$$

(2.19)

in contradiction with (2.18).

Thus, we can conclude that, if $P_i\circ T(x')=P_i\circ T(x'')$ and $t'<t''$, then $T_i(x')<T_i(x'')$, so the proof is complete. $\square$

Now, we exploit the class of admissible deformations $D_L$ in order to construct the solutions $u_k$ and $v_k$. We first construct the solutions $u_k$ (then one can proceed in a similar way to construct the solutions $v_k$). For all $k\in {\mathbb {N}}$, $z\in Z_k$ and $T\in D_L$ with $L\in ]0,1[$, let us set

$$\begin{aligned} E_\psi (k,z,T)=\inf \left\{ E_\psi (u)\,\ u\in H^1_0(T(C^k_z)),\ \int _{T(C^k_z)}|u|^{p+1}dx=1\right\} . \end{aligned}$$

(2.20)

Since $p<\frac{n+2}{n-2}$ when $n\ge 3$, one can easily verify that the infimum in (2.20) is achieved.

Moreover, for all $k\in {\mathbb {N}}$ and $L\in ]0,1[$, also the infimum

$$\begin{aligned} \inf \{E_\psi (k,z,T)\,\ z\in Z_k,\ T\in D_L\} \end{aligned}$$

(2.21)

is achieved, as one can prove by standard arguments using Ascoli-Arzelà Theorem.

For the construction of the functions $u_k$ we need the following Lemmas.

Lemma 2.4

For all $L\in ]0,1[$ we have

$$\begin{aligned} \lim _{k\rightarrow \infty } \min \{E_\psi (k,z,T)\,\ z\in Z_k,\ T\in D_L\}=\infty \end{aligned}$$

(2.22)

and there exists $k(L)\in {\mathbb {N}}$ such that, for all $k\ge k(L)$, $z\in Z_k$ and $T\in D_L$, the infimum

$$\begin{aligned} \inf \left\{ E_\psi (u)\,\ u\in H^1_0(T(C^k_z)),\ \int _{T(C^k_z)}|u|^{p+1}dx<1\right\} \end{aligned}$$

(2.23)

is achieved by a unique minimizing function ${\tilde{u}}^T_{k,z}$. Moreover, we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\sup \left\{ \int _{T(C^k_z)}|{\nabla }{\tilde{u}}^T_{k,z}|^2dx\,\ z\in Z_k,\ T\in D_L\right\} =0. \end{aligned}$$

(2.24)

Proof

For all $k\in {\mathbb {N}}$, let us consider $z^k\in Z_k$ and $T_k\in D_L$ realizing the minimum (2.22) and ${\bar{u}}_k\in H^1_0(T(C^k_{z^k}))$ realizing the minimum $E_\psi (k,z^k,T_k)$.

Let us extend the function ${\bar{u}}_k$ in all of $\Omega$ by the value zero in $\Omega \setminus C^k_{z^k}$. Since $T_k\in D_L$ $\forall k\in {\mathbb {N}}$, taking into account the second inequality in (2.10), we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\textrm{meas}T_k(C^k_{z^k})=0, \end{aligned}$$

(2.25)

so (up to a subsequence) ${\bar{u}}_k\rightarrow 0$ almost everywhere in $\Omega$. It follows that

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _\Omega |{\nabla }{\bar{u}}_k|^2dx=\infty \end{aligned}$$

(2.26)

otherwise, since $p<\frac{n+2}{n-2}$ for $n\ge 3$, ${\bar{u}}_k\rightarrow 0$ also in $L^{p+1}(\Omega )$, which is impossible because $\int _{\Omega }|{\bar{u}}_k|^{p+1}dx=1$ $\forall k\in {\mathbb {N}}$. As a consequence, since

$$\begin{aligned} E_\psi (k,z_k,T^k)=\frac{1}{2} \int _\Omega |{\nabla }{\bar{u}}_k|^2dx-\frac{1}{p+1}-\int _\Omega {\bar{u}}_k\psi \, dx, \end{aligned}$$

(2.27)

we obtain (2.22).

Notice that (2.22) implies that for all $L\in ]0,1[$ there exists $k(L)\in {\mathbb {N}}$ satisfying

$$\begin{aligned}{} & {} 0<\min \left\{ E_\psi (u)\,\ u\in H^1_0(T(C^k_z)),\ \int _{T(C^k_z)}|u|^{p+1}dx=1\right\} \nonumber \\{} & {} \quad \forall k\ge k(L),\ \forall z\in Z_k,\ \forall T\in D_L. \end{aligned}$$

(2.28)

Since $E_\psi (0)=0$, it follows by standard arguments that for all $k\ge k(L)$, $z\in Z_k$ and $T\in D_L$ there exists ${\tilde{u}}^T_{k,z}\in H^1_0(T(C^k_z))$ such that

$$\begin{aligned} E_\psi ( {\tilde{u}}^T_{k,z})=\min \left\{ E_\psi (u)\,\ u\in H^1_0(T(C^k_z)),\ \int _{T(C^k_z)}|u|^{p+1}dx<1\right\} . \end{aligned}$$

(2.29)

Taking into account that

$$\begin{aligned} E_\psi ( {\tilde{u}}^T_{k,z})\le E_\psi (0)=0\quad \forall k\ge k(L),\ \forall z\in Z_k,\ \forall T\in D_L, \end{aligned}$$

(2.30)

it follows that

$$\begin{aligned} \sup \left\{ \int _{T(C^k_z)}|{\nabla }{\tilde{u}}^T_{k,z}|^2dx\,\ k\ge k(L),\ z\in Z_k,\ T\in D_L\right\} <\infty . \end{aligned}$$

(2.31)

In order to prove (2.24), we argue by contradiction and assume that for all $k\ge k(L)$ there exist $z^k\in Z_k$ and $T_k\in D_L$ such that

$$\begin{aligned} \liminf _{k\rightarrow \infty }\int _{T_k(C^k_{z^k})}|{\nabla }{\tilde{u}}^{T_k}_{k,z^k}|^2dx>0. \end{aligned}$$

(2.32)

From (2.31) we infer that the sequence $({\tilde{u}}^{T_k}_{k,z^k})_k$ (with ${\tilde{u}}^{T_k}_{k,z^k}$ extended by the value zero outside $T_k(C^k_{z^k})$) is bounded in $H^1_0(\Omega )$. Moreover, up to a subsequence, ${\tilde{u}}^{T_k}_{k,z^k}\rightarrow 0$ as $k\rightarrow \infty$ almost everywhere in $\Omega$ because $T_k\in D_L$, so $\textrm{meas}(T_k(C^k_{z^k}))\rightarrow 0$ as $k\rightarrow \infty$. Therefore, ${\tilde{u}}^{T_k}_{k,z^k}\rightarrow 0$ as $k\rightarrow \infty$ also in $L^{p+1}(\Omega )$. Then, from $E_\psi ({\tilde{u}}^{T_k}_{k,z^k})\le 0$ $\forall k\in {\mathbb {N}}$ it follows easily that

$$\begin{aligned} \limsup _{k\rightarrow \infty }\int _{T_k(C^k_{z^k})}|{\nabla }{\tilde{u}}^{T_k}_{k,z^k}|^2dx=0 \end{aligned}$$

(2.33)

in contradiction with (2.32). Thus, we can conclude that (2.24) holds.

Finally, notice that the functional $E_\psi$ is strictly convex in a suitable neighborhood of zero. Therefore, for k large enough, ${\tilde{u}}^{T_k}_{k,z^k}$ is the unique minimizing function for (2.23) for all $z\in Z_k$ and $T\in D_L$. So the proof is complete. $\square$

Lemma 2.5

For all $k\ge k(L)$, $z\in Z_k$ and $T\in D_L$, there exists a function $u^{T}_{k,z}$ in $H^1_0(T(C^k_z))$ such that $u^{T}_{k,z}\not \equiv {\tilde{u}}^{T}_{k,z}$, $\sigma (z)[u^{T}_{k,z}-{\tilde{u}}^{T}_{k,z}]\ge 0$ in $T(C^k_z)$ and

$$\begin{aligned} \begin{aligned} E_\psi (u^{T}_{k,z})&=M_\psi (u^{T}_{k,z}) \\&=\min \{M_\psi (u)\,\ u\in H^1_0(T(C^k_z)),\ u\not \equiv {\tilde{u}}^{T}_{k,z},\ \sigma (z)[u -{\tilde{u}}^{T}_{k,z}]\ge 0\ \text{ in } T(C^k_z)\} \end{aligned}\nonumber \\ \end{aligned}$$

(2.34)

where, for all $u\in H^1_0(T(C^k_z))$, $M_\psi (u)$ is defined by

$$\begin{aligned} M_\psi (u)=\max \{E_\psi ({\tilde{u}}^{T}_{k,z}+t(u-{\tilde{u}}^{T}_{k,z}))\,\ t\ge 0\}. \end{aligned}$$

(2.35)

Proof

First notice that the maximum in (2.35) is achieved for all $u\in H^1_0(T(C^k_z))$ because $p>1$. Now, let us consider a sequence $(u_i)_i$ in $H^1_0(T(C^k_z))$ such that $u_i\not \equiv {\tilde{u}}^{T}_{k,z}$, $\sigma (z)[ u_i-{\tilde{u}}^{T}_{k,z} ]\ge 0$ in $T(C^k_z)$ $\forall i\in {\mathbb {N}}$ and

$$\begin{aligned} \lim _{i\rightarrow \infty }M_\psi (u_i)=\inf \{M_\psi (u)\,\ u\in H^1_0(T(C^k_z)),\ u\not \equiv {\tilde{u}}^{T}_{k,z^k},\ \sigma (z)[ u -{\tilde{u}}^{T}_{k,z} ]\ge 0 \text{ in } T(C^k_z)\}.\nonumber \\ \end{aligned}$$

(2.36)

Then, let us set $w_i=\Vert u_i-{\tilde{u}}^{T}_{k,z}\Vert ^{-1}_{L^{p+1}}( u_i-{\tilde{u}}^{T}_{k,z})$ and notice that, obviously, $M_\psi (u_i)=M_\psi ({\tilde{u}}^{T}_{k,z}+w_i)$. Moreover notice that, since the sequence $(w_i)_i$ is bounded in $L^{p+1}$, (2.36) implies that it is bounded also in $H^1_0$. Since $p<\frac{n+2}{n-2}$ when $n\ge 3$, it follows that (up to a subsequence) $(w_i)_i$ converges weakly in $H^1_0$, in $L^{p+1}$ and almost everywhere to a function ${\hat{w}}\in H^1_0(T(C^k_z))$. As a consequence, $\Vert {\hat{w}}\Vert _{L^{p+1}}=1$ and $\sigma (z){\hat{w}}\ge 0$ in $T(C^k_z)$. Indeed, $w_i\rightarrow {\hat{w}}$ as $i\rightarrow \infty$ strongly in $H^1_0(T(C^k_z))$. In fact, since we have the weak convergence, arguing by contradiction assume that $\Vert w_i\Vert ^2_{H^1_0}$ does not converge to $\Vert {\hat{w}}\Vert ^2_{H^1_0}$ as $i\rightarrow \infty$, that is

$$\begin{aligned} \int _{T(C^k_z)}|{\nabla }{\hat{w}}|^2dx<\lim _{i\rightarrow \infty } \int _{T(C^k_z)}|{\nabla }w_i|^2dx, \end{aligned}$$

(2.37)

which, combined with the weak convergence, implies $M_\psi ({\tilde{u}}+{\hat{w}})<\lim _{i\rightarrow \infty }M_\psi ({\tilde{u}}+w_i)$. Therefore, we obtain a contradiction because ${\hat{w}}\not \equiv 0$ and, as a consequence, $\lim _{i\rightarrow \infty } M_\psi ({\tilde{u}}+w_i)\le M_\psi ({\tilde{u}}+{\hat{w}})$ because of (2.36). Thus, we can conclude that $w_i\rightarrow {\hat{w}}$ in $H^1_0( T(C^k_z))$ as $i\rightarrow \infty$, which imples $\lim _{i\rightarrow \infty }M_\psi ({\tilde{u}}+w_i)=M_\psi ({\tilde{u}}+{\hat{w}})$.

Moreover, since $p>1$, there exists ${\hat{t}}>0$ such that $E_\psi ({\tilde{u}}^T_{k,z}+{\hat{t}}{\hat{w}})=M_\psi ({\tilde{u}}^T_{k,z}+{\hat{t}}{\hat{w}})$, so all the assertions in Lemma 2.5 hold with $u^T_{k,z}={\tilde{u}}^T_{k,z}+{\hat{t}}{\hat{w}}$. $\square$

Remark 2.6

Notice that the function $u^T_{k,z}$ given by Lemma 2.5, for k large enough, satisfies $E_\psi (u^T_{k,z})\ge E_\psi (k,z,T)$ because $u^T_{k,z}\not \equiv {\tilde{u}}^T_{k,z}$ in $T(C^k_z)$.

Thus, by (2.22) we get

$$\begin{aligned} \lim _{k\rightarrow \infty }\min \{E_\psi (u^T_{k,z})\,\ z\in Z_k,\ T\in D_L\}=\infty . \end{aligned}$$

(2.38)

Now, we extend every function $u^T_{k,z}$ in all of $\Omega$ by the value zero outside $T(C^k_z)$ and we consider the function $u^T_k\in H^1_0(\Omega )$ defined by $u^T_k=\sum _{z\in Z_k}u^T_{k,z}$. Using Ascoli–Arzelà Theorem, one can verify that for all $k\ge k(L)$ there exists an admissible deformation $T^L_k\in D_L$ such that

$$\begin{aligned} E_\psi \left( u^{T^L_k}_k\right) =\min \{E_\psi (u^T_k)\,\ T\in D_L\}. \end{aligned}$$

(2.39)

In next section we show that $u^{T^L_k}_k$ is a solution of problem (1.1) for k large enough and that Proposition 2.2 holds with $u_k=u^{T^L_k}_k$ and $T_{k,u}=T^L_k$. In order to construct the solutions $v_k$, we proceed in analogous way. In fact, as in Lemma 2.5, for all $k\ge k(L)$, $z\in Z_k$ and $T\in D_L$, there exists also a function $v^T_{k,z}$ in $H^1_0(T(C^k_z))$ such that $v^T_{k,z}\ne {\tilde{u}}^T_{k,z}$, $\sigma (z)[v^T_{k,z}-{\tilde{u}}^T_{k,z}]\le 0$ in $T(C^k_z)$ and

$$\begin{aligned}{} & {} E_\psi (v^T_{k,z})=M_\psi (v^T_{k,z})=\min \{M_\psi (v)\,\ v\in H^1_0(T(C^k_z)),\ v\not \equiv {\tilde{u}}^T_{k,z},\nonumber \\{} & {} \sigma (z)[v -{\tilde{u}}^T_{k,z}]\le 0 \text{ in } T(C^k_z)\}. \end{aligned}$$

(2.40)

Then, we set $v^T_k=\sum _{z\in Z_k}v^T_{k,z}$ (where $v^T_{k,z}$ is extended in $\Omega$ by the value zero outside $T(C^k_z)$) and, using Ascoli–Arzelà Theorem, we minimize $E_\psi (v^T_k)$ with respect to T in $D_L$. If $T_{k,v}\in D_L$ is a minimizing admissible deformation, the function $v_k^{T_{k,v}}$ is a solution of problem (1.1) for k large enough and Proposition 2.2 holds with $v_k=v_k^{T_{k,v}}$, as we show in next section.

3 Asymptotic estimates and proof of the main results

In this section we describe the asymptotic behaviour as $k\rightarrow \infty$ of the functions $u_k$ and $v_k$, arising in Proposition 2.2, we constructed in Sect. 2. Then, we show that these functions are solutions of problem (1.1) for k large enough and satisfy all the assertions of Proposition 2.2.

As follows from Proposition 2.3, for all $T\in D_L$, $i\in \{1,\ldots ,n\}$ and $t\in [0,1]$, the set $T(F^t_i)$ is the graph of a function $f^{t,T}_i:F^0_i\rightarrow {\mathbb {R}}$ and

$$\begin{aligned} T(C^k_z)=\{x\in \Omega \,\ f^{\frac{z_i}{k},T}_i\circ P_i(x)<x_i<f_i^{\frac{z_i+1}{k},T}\circ P_i(x) \text{ for } i=1,\ldots ,n\}\quad \forall k\in {\mathbb {N}},\ \forall z\in Z_k.\nonumber \\ \end{aligned}$$

(3.1)

In next lemma we prove that $f^{t,T}_i$ is a Lipschitz continuous function.

Lemma 3.1

If $T\in D_L$ with $L\in ]0,1[$, then for all $i\in \{1,\ldots ,n\}$ and $t\in [0,1]$ we have

$$\begin{aligned} |f^{t,T}_i(x)-f^{t,T}_i(y)|\le \frac{L}{1-L}\, |x-y|\qquad \forall x,y\in F^0_i. \end{aligned}$$

(3.2)

Proof

For all x, y in $F^0_i$, there exist $x^t,y^t$ in $F^t_i$ such that $P_i\circ T(x^t)=x$, $P_i\circ T(y^t)=y$ and, as a consequence, $f^{t,T}_i(x)=T_i(x^t)$, $f^{t,T}_i(y)=T_i(y^t)$.

Thus, since $x^t_i=y^t_i=t$ and $T\in D_L$, we obtain

$$\begin{aligned} \begin{aligned} |f^{t,T}_i(x)- f^{t,T}_i(y)|&=|T_i(x^t)-T_i(y^t)|=|T_i(x^t)-T_i(y^t)-x^t_i+y^t_i|\\&\le |T(x^t)-T(y^t)-x^t+y^t|\\&\le L|x^t-y^t|. \end{aligned} \end{aligned}$$

(3.3)

Moreover, since $L\in ]0,1[$, we obtain

$$\begin{aligned} \begin{aligned} |x-y|&=|P_i\circ T(x^t)- P_i\circ T(y^t)|=|P_i[T(x^t)-T(y^t)+y^t-x^t-(y^t-x^t)]|\\&\ge |x^t-y^t|-|P_i[T(x^t)-T(y^t)+y^t-x^t ]|\\&\ge (1-L)|x^t-y^t| \end{aligned} \end{aligned}$$

(3.4)

which, combined with (3.3), implies (3.2). $\square$

Let us denote by $\textrm{Lip}(f^{t,T}_i)$ the best Lipschitz constant of the function $f^{t,T}_i$, that is

$$\begin{aligned} \textrm{Lip}(f^{t,T}_i)=\sup \left\{ \frac{|f^{t,T}_i(x)-f^{t,T}_i(y)|}{|x-y|}\,\ x,y\in P_i(\Omega ),\ x\ne y\right\} . \end{aligned}$$

(3.5)

Then, from (3.2) it follows that $\textrm{Lip}(f^{t,T}_i)\rightarrow 0$ as $L\rightarrow 0$.

Corollary 3.3 shows, in some sense, that also the converse in true. Notice that, if we set $S_T(x)=T(x)-x$ $\forall x\in \Omega$, then $T\in D_L$ if and only if

$$\begin{aligned} \textrm{Lip}(S_T):=\sup \left\{ \frac{|S_T(x)-S_T(y)|}{|x-y|}\,\ x,y\in \Omega ,\ x\ne y\right\} \le L. \end{aligned}$$

(3.6)

Moreover, it is obvious that the set $D_L$ may be also written as

$$\begin{aligned} D_L=\{T:\overline{\Omega }\rightarrow \overline{\Omega }\,\ T(\overline{\Omega })=\overline{\Omega },\ T(F^t_i)=F^t_i \text{ for } t=0,1,\ i=1,\ldots ,n,\ \textrm{Lip}(S_T)\le L\}.\nonumber \\ \end{aligned}$$

(3.7)

Lemma 3.2

Let $T\in D_L$ with $L\in ]0,1[$ and assume that there exists $\Lambda \in \left]0,\frac{1}{n}\right[$ such that

$$\begin{aligned} \textrm{Lip}(F^{t,T}_i)\le \Lambda \qquad \forall t\in [0,1],\ \forall i\in \{1,\ldots ,n\} \end{aligned}$$

(3.8)

and

$$\begin{aligned} |f^{t_1,T}_i(x)-f^{t_2,T}_i(x)+t_2-t_1|\le \Lambda |t_1-t_2|\qquad \forall t_1,t_2\in [0,1],\ \forall i\in \{1,\ldots ,n\},\ \forall x\in F^0_i.\nonumber \\ \end{aligned}$$

(3.9)

Then,

$$\begin{aligned} |T(x)-T(y)-x+y|\le \frac{(n+1)\sqrt{n} \, \Lambda }{1-n\Lambda }\, |x-y|\quad \forall x,y\in \overline{\Omega }, \end{aligned}$$

(3.10)

that is $\textrm{Lip}(S_T)\le \frac{(n+1)\sqrt{n} \,\Lambda }{1-n\Lambda }$, so $T\in D_{L(\Lambda )}$ with $L(\Lambda )=\frac{(n+1)\sqrt{n} \, \Lambda }{1-n\Lambda }$.

Proof

Notice that $T_i(x)=f_i^{x_i,T}(P_i\circ T(x))$ for all $x\in \overline{\Omega }$ and $i\in \{1,\ldots ,n\}$.

Thus, for $x,y\in \Omega$ and $h=y-x$, we obtain

$$\begin{aligned} \begin{aligned} T_i(x+h)-T_i(x)-h_i&= f^{x_i+h_i,T}_i(P_i\circ T(x+h))- f^{x_i,T}_i(P_i\circ T(x+h))-h_i\\&\quad \quad +f^{x_i,T}_i(P_i\circ T(x+h))-f^{x_i,T}_i(P_i\circ T(x))\\&=\mu _ih_i+\sum _{j=1}^n\nu _i^j[T_j(x+h)-T_j(x)] \end{aligned} \end{aligned}$$

(3.11)

where, for all i and j in $\{1,\ldots ,n\}$, $\mu _i$ and $\nu _i^j$ are suitable numbers in $[-\Lambda ,\Lambda ]$ because of our assumptions on the functions $f^{t,T}_i$.

It follows that

$$\begin{aligned} |T_i(x+h)-T_i(x)-h_i|{} & {} \le \Lambda |h_i|+\Lambda \sum _{j=1}^n |T_j(x+h)-T_j(x)-h_j|\nonumber \\{} & {} \quad +\Lambda \sum _{j=1}^n|h_j|\quad \forall i\in \{1,\ldots ,n\} \end{aligned}$$

(3.12)

and, summing up,

$$\begin{aligned} \sum _{i=1}^n|T_i(x+h)-T_i(x)-h_i|\le (1+n)\Lambda \sum _{i=1}^n|h_i|+n\Lambda \sum _{i=1}^n|T_i(x+h)-T_i(x)-h_i|. \nonumber \\ \end{aligned}$$

(3.13)

Since $\Lambda <\frac{1}{n}$, we obtain

$$\begin{aligned} \sum _{i=1}^n|T_i(x+h)-T_i(x)-h_i|\le \frac{(n+1)\Lambda }{1-n\Lambda }\, \sum _{i=1}^n|h_i| \end{aligned}$$

(3.14)

which implies

$$\begin{aligned}{} & {} |T(x+h)-T(x)-h|\le \sum _{i=1}^n|T_i(x+h)-T_i(x)-h_i|\nonumber \\{} & {} \quad \le \frac{(n+1)\Lambda }{1-n\Lambda }\,\sum _{i=1}^n|h_i|\le \frac{(n+1)\sqrt{n}\,\Lambda }{1-n\Lambda }|h|. \end{aligned}$$

(3.15)

So the proof is complete. $\square$

The following corollary is a direct consequence of Lemma 3.2.

Corollary 3.3

Let $(T_k)_k$ be a sequence in $D_L$ with $L\in ]0,1[$ and assume that, for a suitable sequence $(\Lambda _k)_k$ in $\left]0,\frac{1}{n} \right[$, the same conditions as in Lemma 3.2 are satisfied with T replaced by $T_k$ and $\Lambda$ by $\Lambda _k$ for all $k\in {\mathbb {N}}$.

Then, $\lim _{k\rightarrow \infty }\Lambda _k=0$ implies $\lim _{k\rightarrow \infty }\textrm{Lip}(S_{T_k})=0$.

Remark 3.4

Notice that, if $\textrm{Lip}(S_{T_k})\mathop {\longrightarrow }0$ as $k\rightarrow \infty$, then $S_{T_k}$ converges to a constant function $S_\infty$ uniformly in $\Omega$. Moreover, taking into account that $T_k\in D_L$ $\forall k\in {\mathbb {N}}$ so $T_k$ must satisfy suitable conditions on $\partial \Omega$, we can say that $S_\infty \equiv 0$, that is $T_k$ converges to the identity function in $\Omega$.

Now, let us prove the assertions of Proposition 2.2 for the function $u_k=u^{T^L_k}_k$ (in a similar way one can proceed for the function $v_k=v^{T_k,v}_k$). First, we prove the following proposition (here we use the notation introduced in Lemmas 2.4 and 2.5).

Proposition 3.5

For all $k\ge k(L)$ the function $u_k=u^{T^L_k}_k$ (extended to ${\mathbb {R}}^n$ by the value zero in ${\mathbb {R}}^n\setminus \Omega$) has the following asymptotic behaviour.

For every choice of $z^k$ in $Z_k$, there exists a function ${\hat{T}}:\overline{\Omega }\rightarrow {\mathbb {R}}^n$ such that

$$\begin{aligned} {\hat{T}}(0)=0,\qquad (1-L)|x-y|\le |{\hat{T}}(x)-{\hat{T}}(y)|\le (1+L)|x-y|\quad \forall x,y\in \overline{\Omega }\qquad \quad \end{aligned}$$

(3.16)

and, if we set $\chi :={\hat{T}}(\Omega )$, the function $U_{z^k}$ defined by

$$\begin{aligned} {U}_{z^k}(x)=\sigma (z^k)k^{-\frac{2}{p-1}}u^{T^L_k}_k\left( \frac{x}{k}+T^L_k\left( \frac{z^k}{k}\right) \, \right) \qquad \forall x\in {\mathbb {R}}^n, \end{aligned}$$

(3.17)

restricted to $\chi$, as $k\rightarrow \infty$ converges in $H^1(\chi )$ to a positive solution $U_\chi$ of the Dirichlet problem

$$\begin{aligned} -\Delta U=|U|^{p-1}U\quad \text{ in } \chi ,\qquad U=0\quad \text{ on } \partial \chi , \end{aligned}$$

(3.18)

satisfying

$$\begin{aligned} \left( \int _\chi |U_\chi |^{p+1}dx\right) ^{-\frac{2}{p-1}}\int _\chi |{\nabla }U_\chi |^2dx=m(\chi ), \end{aligned}$$

(3.19)

where

$$\begin{aligned} m(\chi ):=\min \left\{ \int _\chi |{\nabla }U|^2dx\,\ U\in H^1_0(\chi ),\ \int _\chi |U|^{p+1}dx=1\right\} . \end{aligned}$$

(3.20)

Moreover, we have

$$\begin{aligned} \begin{aligned} \lim _{k\rightarrow \infty }k^{\left( n-2\frac{p+1}{p-1}\right) }E_\psi (u_{k,z^k}^{T_k^L})&=\lim _{k\rightarrow \infty }\left[ \frac{1}{2} \int _\chi |{\nabla }U_{z^k}|^2dx-\frac{1}{p+1}\int _\chi |U_{z^k}|^{p+1}dx\right] \\&=\frac{1}{2}\int _\chi |{\nabla }U_\chi |^2dx-\frac{1}{p+1}\int _\chi |U_\chi |^{p+1}dx\\&=\left( \frac{1}{2}-\frac{1}{p+1}\right) [m(\chi )]^{\frac{p+1}{p-1}}. \end{aligned} \end{aligned}$$

(3.21)

Proof

For all $k\in {\mathbb {N}}$, let us rescale problem (1.1) by replacing every function $u\in H^1_0(\Omega )$ by the function $R^ku\in H^1_0(k\Omega )$ defined by

$$\begin{aligned} R^ku(x)=k^{-\frac{2}{p-1}}u\left( \frac{x}{k}\right) \qquad \forall x\in k\Omega \end{aligned}$$

(3.22)

(here $R^ku$ is extended by the value zero outside $k\Omega$).

Then, our problem becomes

$$\begin{aligned} -\Delta U=|U|^{p-1}U+\psi _k\quad \text{ in } k\Omega ,\qquad U=0\quad \text{ on } \partial (k\Omega ) \end{aligned}$$

(3.23)

where $\psi _k\in L^2(k\Omega )$ is defined by

$$\begin{aligned} \psi _k(x)=k^{-\frac{2p}{p-1}}\psi \left( \frac{x}{k}\right) \qquad \forall x\in k\Omega . \end{aligned}$$

(3.24)

Moreover, the corresponding functional becomes

$$\begin{aligned} E^k(U)=\frac{1}{2} \int _{k\Omega }|{\nabla }U|^2dx-\frac{1}{p+1}\int _{k\Omega }|U|^{p+1}dx-\int _{k\Omega }\psi _kU\, dx, \end{aligned}$$

(3.25)

defined for all $u\in H^1_0(k\Omega )$.

Since $T^L_k\in D_L$ $\forall k\in {\mathbb {N}}$, so in particular it satisfies (2.10), also the function $kT^L_k\left( \frac{x+z^k}{k}\right)$, defined for all $x\in \overline{\Omega }$, satisfies (2.10) and, as a consequence,

$$\begin{aligned} kT^L_k\left( \frac{x+z^k}{k}\right) \in B\left( kT^L_k\left( \frac{z^k}{k}\right) ,(1+L)\sqrt{n}\right) \qquad \forall x\in \overline{\Omega }. \end{aligned}$$

(3.26)

Therefore, using Ascoli–Arzelà Theorem, we infer that (up to a subsequence) the function $kT^L_k\left( \frac{\cdot \, +z^k}{k}\right) -kT^L_k\left( \frac{z^k}{k}\right)$ converges as $k\rightarrow \infty$ to a function ${\hat{T}}:\overline{\Omega }\mathop {\longrightarrow }B(0,(1+L)\sqrt{n})$ uniformly in $\overline{\Omega }$. As a consequence, ${\hat{T}}$ satisfies (3.16) in $\overline{\Omega }$.

From Lemmas 2.4 and 2.5 we infer that $R^k{\tilde{u}}^{T^L_k}_{k,z^k}$ and $R^k u^{T^L_k}_{k,z^k}$ belong to $H^1_0(kC^k_{z^k})$ and that

$$\begin{aligned}{} & {} E^k( R^k{\tilde{u}}^{T^L_k}_{k,z^k})=\min \left\{ E^k(U)\,\ U\in H^1_0(kT^L_k(C^k_{z^k})),\ \int _{ kT^L_k(C^k_{z^k})}|U|^{p+1} <1\right\} \qquad \forall k\in {\mathbb {N}}, \end{aligned}$$

(3.27)

$$\begin{aligned}{} & {} \lim _{k\rightarrow \infty } E^k(R^k {\tilde{u}}^{T^L_k}_{k,z^k})=\lim _{k\rightarrow \infty } \int _{ kT^L_k(C^k_{z^k})}|{\nabla }R^k{\tilde{u}}^{T^L_k}_{k,z^k}|^2dx=0. \end{aligned}$$

(3.28)

Moreover,

$$\begin{aligned} \begin{aligned}&E^k( R^k u^{T^L_k}_{k,z^k})=M^k( R^k u^{T^L_k}_{k,z^k}) =\min \{M^k(U)\,\ U\in H^1_0(kT^L_k(C^k_{z^k})),\ U\not \equiv R^k {\tilde{u}}^{T^L_k}_{k,z^k},\\&\sigma (z^k)[U-R^k {\tilde{u}}^{T^L_k}_{k,z^k}]\ge 0 \text{ in } kT^L_k(C^k_{z^k})\} \end{aligned}\qquad \end{aligned}$$

(3.29)

where $M^k(U)$ is defined by

$$\begin{aligned} M^k(U)=\max \{ E^k(R^k {\tilde{u}}^{T^L_k}_{k,z^k}+t(U-R^k {\tilde{u}}^{T^L_k}_{k,z^k})\,\ t\ge 0\}\qquad \forall U\in H^1_0( kT^L_k(C^k_{z^k})). \qquad \end{aligned}$$

(3.30)

Notice that

$$\begin{aligned} \int _{k\Omega }\psi ^2_kdx=k^{n-\frac{4p}{p-1}}\int _\Omega \psi ^2dx \end{aligned}$$

(3.31)

where $n<\frac{4p}{p-1}$ under our assumptions on p. In fact, for $n\le 4$ it is obviously true because $p>1$ while for $n>4$ it is true because $1<p<\frac{n+2}{n-2}$, as one can easily verify by direct computation (taking into account that $\frac{n+2}{n-2}<\frac{n}{n-4}$). As a consequence, we obtain in particular

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{kT^L_k(C^k_{z^k})}\psi _k^2dx=0. \end{aligned}$$

(3.32)

Therefore, we infer that the function $U_{z^k}$ satisfies all the assertions in Proposition 3.5, that is its restriction to $\chi$ converges to a positive solution $U_\chi$ of the asymptotic problem (3.18), satisfying the minimality condition (3.19).

In fact, (3.28) and (3.32) imply

$$\begin{aligned} \begin{aligned} 0&<\liminf _{k\rightarrow \infty }\left[ \frac{1}{2}\int _\chi |{\nabla }U_{z^k}|^2dx-\frac{1}{p+1}\int _{\chi }|U_{z^k}|^{p+1}dx\right] \\&\le \limsup _{k\rightarrow \infty }\left[ \frac{1}{2}\int _\chi |{\nabla }U_{z^k}|^2dx-\frac{1}{p+1}\int _{\chi }|U_{z^k}|^{p+1}dx\right] <\infty \end{aligned} \end{aligned}$$

(3.33)

and

$$\begin{aligned} \lim _{k\rightarrow \infty }\left[ \int _\chi |{\nabla }U_{z^k}|^2dx- \int _{\chi }|U_{z^k}|^{p+1}dx\right] =0. \end{aligned}$$

(3.34)

It follows that

$$\begin{aligned} \limsup _{k\rightarrow \infty }\frac{\int _\chi |{\nabla }U_{z^k}|^2dx}{\left( \int _\chi |U_{z^k}|^{p+1}dx\right) ^{\frac{2}{p+1}}} <\infty \end{aligned}$$

(3.35)

so, up to a subsequence, $\left( \int _\chi |U_{z^k}|^{p+1}dx\right) ^{-\frac{1}{p+1}}U_{z^k}$ converges as $k\rightarrow \infty$ to a positive function ${\overline{U}}_\chi \in H^1_0(\chi )$ almost everywhere in $\chi$, strongly in $L^{p+1}(\chi )$ and weakly in $H^1(\chi )$.

Moreover, the minimality property of $u^{T^L_k}_{k,z^k}$ implies, by standard arguments, that

$$\begin{aligned} \int _\chi |{\nabla }{\overline{U}}_\chi |^2dx=m(\chi ) \end{aligned}$$

(3.36)

and that, as $k\rightarrow \infty$, $U_{z^k}$ converges strongly in $H^1(\chi )$ to the function $U_\chi =m(\chi )^{\frac{1}{p-1}}{\overline{U}}_\chi$, which is a positive solution of problem (3.18).

Therefore, taking also into account (3.24), we obtain

$$\begin{aligned} \begin{aligned} \lim _{k\rightarrow \infty }k^{\left( n-2\frac{p+1}{p-1}\right) }E_\psi \left( u^{T^L_k}_{k,z^k}\right)&=\lim _{k\rightarrow \infty }E^k\left( R^k u^{T^L_k}_{k,z^k}\right) \\&= \lim _{k\rightarrow \infty }\left[ \frac{1}{2}\int _\chi |{\nabla }U_{z^k}|^2dx-\frac{1}{p+1}\int _\chi |U_{z^k}|^{p+1}dx\right] \\&=\left( \frac{1}{2} -\frac{1}{p+1}\right) [m(\chi )]^{\frac{p+1}{p-1}}. \end{aligned}\qquad \end{aligned}$$

(3.37)

So the proof is complete. $\square$

In next lemma we describe other properties of the function ${\hat{T}}$ and of the domain ${\hat{T}}(\Omega )$ arising in Proposition 3.5.

Lemma 3.6

Let $(z^k)_k$, ${\hat{T}}$ and $\chi$ be as in Proposition 3.5. Then, the function $S_{{\hat{T}}}:\overline{\Omega }\rightarrow {\mathbb {R}}^n$ defined by $S_{{\hat{T}}}(x)={\hat{T}}(x)-x$ $\forall x\in \overline{\Omega }$, satisfies the Lipschitz condition

$$\begin{aligned} |S_{{\hat{T}}}(x)-S_{{\hat{T}}}(y)|\le L |x-y|\qquad \forall x,y\in \overline{\Omega }. \end{aligned}$$

(3.38)

Moreover, for every $i\in \{1,\ldots ,n\}$ there exist two functions $f^0_i,f^1_i:P_i(\chi )\rightarrow {\mathbb {R}}$, Lipschitz continuous with Lipschitz constant $\frac{L}{1-L}$, such that $f^0_i\circ P_i(0)=0$, $f^0_i\circ P_i(x)<f^1_i\circ P_i(x)$ $\forall x\in \chi$ and

$$\begin{aligned} \chi =\{x\in {\mathbb {R}}^n\,\ P_i(x)\in P_i(\chi ),\ f^0_i\circ P_i(x)<x_i<f^1_i\circ P_i(x)\ \text{ for } i=1,\ldots ,n\}. \qquad \end{aligned}$$

(3.39)

Proof

Notice that, as the functions $S_{T^L_k}:\overline{\Omega }\rightarrow {\mathbb {R}}^n$ defined by $S_{T^L_k}(x)=T^L_k(x)-x$ $\forall x\in \Omega$, also the functions $k\left[ T^L_k\left( \frac{x+z^k}{k}\right) -T^L_k\left( \frac{z^k}{k}\right) -\frac{x}{k}\right]$ are Lipschitz continuous with Lipschitz constant L for all $k\in {\mathbb {N}}$. Therefore, as $k\rightarrow \infty$, we infer that the function $S_{{\hat{T}}}$ satisfies (3.38). In order to obtain the functions $f^0_i$ and $f^1_i$, we use Lemma 3.1. From (3.1) it follows that

$$\begin{aligned} T^L_k(C^k_{z^k})=\{x\in \Omega \, \ f_i^{\frac{z^k_i}{k},T^L_k}\circ P_i(x)<x_i<f_i^{\frac{z^k_i+1}{k},T^L_k}\circ P_i(x)\ \text{ for } i=1,\ldots ,n\}. \qquad \end{aligned}$$

(3.40)

Now, notice that, as the functions $f_i^{\frac{z^k_i}{k},T^L_k}$ and $f_i^{\frac{z^k_i+1}{k},T^L_k}$, also the functions $f^0_{i,k}$ and $f^1_{i,k}$ defined by

$$\begin{aligned} f^0_{i,k}(x)=k \left[ f_i^{\frac{z^k_i}{k},T^L_k}\left( \frac{x}{k}+P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right) -f_i^{\frac{z^k_i}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right] \end{aligned}$$

(3.41)

and

$$\begin{aligned} f^1_{i,k}(x)=k\left[ f_i^{\frac{z^k_i+1}{k},T^L_k}\left( \frac{x}{k}+P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right) -f_i^{\frac{z^k_i}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right] \end{aligned}$$

(3.42)

are both Lipschitz continuous with Lipschitz constant $\frac{L}{1-L}$. Moreover, for all $k\in {\mathbb {N}}$ we have

$$\begin{aligned} f^0_{i,k}(0)=0 \quad \forall i\in \{1,\ldots ,n\}, \qquad k\left| T^L_k\left( \frac{x+z^k}{k}\right) -T^L_k\left( \frac{z^k}{k}\right) \right| \le (1+L)\sqrt{n} \quad \forall x\in \overline{\Omega }\nonumber \\ \end{aligned}$$

(3.43)

and

$$\begin{aligned}{} & {} f^1_{i,k}(x)-f^0_{i,k}(x)\ge (1-L)|x^1_{i,k}-x^0_{i,k}|\ge (1-L)>0 \quad \forall x\in P_i(\overline{\chi }),\nonumber \\{} & {} \quad \forall i\in \{1,\ldots ,n\},\ \forall k\in {\mathbb {N}}, \end{aligned}$$

(3.44)

where $x^1_{i,k}$ and $x^0_{i,k}$ are the points in $\overline{\Omega }$ such that

$$\begin{aligned} k\left[ T^L_k\left( \frac{x^1_{i,k}+z^k}{k}\right) -T^L_k\left( \frac{z^k}{k}\right) \right] =(x,f^1_{i,k}(x)) \end{aligned}$$

(3.45)

and

$$\begin{aligned} k\left[ T^L_k\left( \frac{x^0_{i,k}+z^k}{k}\right) -T^L_k\left( \frac{z^k}{k}\right) \right] =(x,f^0_{i,k}(x)), \end{aligned}$$

(3.46)

which implies $|x^1_{i,k}-x^0_{i,k}|\ge 1$.

Therefore, by Ascoli–Arzelà Theorem we can say that, up to a subsequence, the functions $f^1_{i,k}$ and $f^0_{i,k}$ converge as $k\rightarrow \infty$ uniformly in $P_i(\overline{\chi })$ respectively to functions $f^1_i$ and $f^0_i$ satisfying all the assertions in Lemma 3.6. $\square$

Lemma 3.7

Let $(z^k)_k$ and $\chi$ be as in Proposition 3.5. Then, for every choice of $z^k$ in $Z_k$, the domain $\chi$ is a cube of ${\mathbb {R}}^n$ having a vertex in the origin and the sides of lenght 1. Moreover, we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\max \{\big |k^n|T^L_k(C^k_z)|-1\big |\,\ z\in Z_k\}=0 \end{aligned}$$

(3.47)

(where $|T^L_k(C^k_z)|$ denotes the volume of $T^L_k(C^k_z)$) and

$$\begin{aligned} \lim _{k\rightarrow \infty }\max \left\{ \left| k^{\left( n-2\,\frac{p+1}{p-1}\right) }E_\psi (u^{T^L_k}_{k,z})-\left( \frac{1}{2} -\frac{1}{p+1}\right) {\bar{m}}^{\frac{p+1}{p-1}}\right| \,\ z\in Z_k\right\} =0 \end{aligned}$$

(3.48)

where ${\bar{m}}=m(\Omega )$.

Proof

Notice that, as we pointed out in the proof of Proposition 3.5, the effect of the term $\psi$ in problem (1.1) tends to vanish as $k\rightarrow \infty$ because in the rescaled problem (3.23) $\psi$ is replaced by the function $\psi _k$ defined by (3.31) and, since $n<\frac{4p}{p-1}$,

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{k\Omega }\psi ^2_kdx=\lim _{k\rightarrow \infty }k^{n-\frac{4p}{p-1}}\int _\Omega \psi ^2dx=0. \end{aligned}$$

(3.49)

As a consequence, taking into account the minimality of $T^L_k$, the interfaces between the domains $kT^L_k(C^k_z)$, with $z\in Z_k$, tend to be flat, so these domains tend as $k\rightarrow \infty$ to polyhedra with 2n faces, having minimality properties inherited by the analogous properties of the domains $kT^L_k(C^k_z)$, related to the minimality of $T^L_k$.

In particular, arguing as in the proof of Proposition 3.5, one can show in addition that, for every $\rho >0$, the function $k\left[ T^L_k\left( \frac{x+z^k}{k}\right) -T^L_k\left( \frac{z^k}{k}\right) \right]$ (up to a subsequence) converges as $k\rightarrow \infty$ to a function ${\hat{T}}$ uniformly in the domain $\Omega _\rho =\cup _{z\in {\mathbb {Z}}_\rho }{\overline{C}}^1_z$, where

$$\begin{aligned} {\mathbb {Z}}_\rho =\{z\in {\mathbb {Z}}^n\,\ |z|<\rho ,\ z^k+z\in Z_k\ \forall k\in {\mathbb {N}}\} \end{aligned}$$

(3.50)

(not only in $\overline{\Omega }$, which is strictly enclosed in $\Omega _\rho$ for $\rho >\sqrt{n}$).

Moreover, if we set $\chi _z:={\hat{T}}(C^1_z)$ $\forall z\in {\mathbb {Z}}_\rho$, the number $\sum _{z\in {\mathbb {Z}}_\rho }[m(\chi _z)]^{\frac{p+1}{p-1}}$ has to be as small as possible for all $\rho >0$. By symmetry reasons, among this polyhedra $\chi$, the cubes of ${\mathbb {R}}^n$ are the unique minimizers of the value ${\varphi }(\chi ):=m(\chi )|\chi |^{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) }$ (where $|\chi |$ is the volume of $\chi$ and $m(\chi )$ is defined in (3.20)) that is, if we set ${\bar{{\varphi }}}={\varphi }(\Omega )$, ${\varphi }(\chi )={\bar{{\varphi }}}$ if $\chi$ is a cube and ${\varphi }(\chi )>{\bar{{\varphi }}}$ otherwise (notice that ${\varphi }(\chi )$ depends only on the shape of $\chi$ and not on its size because it is invariant with respect to translations and rescaling of $\chi$).

By Proposition 3.5, for every choice of $z^k$ in $Z_k$, the corresponding limit domain $\chi$ satisfies

$$\begin{aligned} \lim _{k\rightarrow \infty }k^n|T^L_k(C^k_{z^k})|=|\chi | \end{aligned}$$

(3.51)

and

$$\begin{aligned} \begin{aligned}&\lim _{k\rightarrow \infty }k^{n-2\frac{p+1}{p-1}}E_\psi (u^{T^L_k}_{k,z^k})[k^n|T^L_k(C^k_{z^k})|] ^{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}}\\&\quad =\left( \frac{1}{2}-\frac{1}{p+1}\right) [m(\chi )]^{\frac{p+1}{p-1}}|\chi |^{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}}\\&\quad =\left( \frac{1}{2}-\frac{1}{p+1}\right) [{\varphi }(\chi )]^{\frac{p+1}{p-1}}, \end{aligned} \end{aligned}$$

(3.52)

which implies

$$\begin{aligned} \lim _{k\rightarrow \infty }k^{\left( n-2\frac{p+1}{p-1}\right) }E_\psi (u^{T^L_k}_{k,z^k})=\left( \frac{1}{2}-\frac{1}{p+1}\right) [{\varphi }(\chi )]^{\frac{p+1}{p-1}}|\chi |^{2\left( \frac{1}{2^*}-\frac{1}{p+1}\right) \frac{p+1}{p-1}}. \end{aligned}$$

(3.53)

Therefore, taking into account the minimality of $T^L_k$, it follows that, for every choice of $z^k$ in $Z_k$, the limit domain $\chi$ is a cube, that is ${\varphi }(\chi )={\bar{{\varphi }}}$. As a consequence, since it is true for every choice of $z^k$ in $Z_k$, we can say that

$$\begin{aligned} \lim _{k\rightarrow \infty }\quad \max \left\{ \left| k^{n-2\frac{p+1}{p-1}}E_\psi (u^{T^L_k}_{k,z})[k^n|T^L_k(C^k_z)|]^ {2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}}-\left( \frac{1}{2} -\frac{1}{p+1}\right) {\bar{{\varphi }}}^{\frac{p+1}{p-1}}\right| \, z\in Z_k\right\} =0.\nonumber \\ \end{aligned}$$

(3.54)

Now, let us set

$$\begin{aligned} {\varphi }_k(z)=\left[ k^{n-2\frac{p+1}{p-1}}\left( \frac{1}{2}-\frac{1}{p+1}\right) ^{-1}E_\psi (u^{T^L_k}_{k,z})\right] ^{\frac{p-1}{p+1}}[k^n|T^L_k(C^k_z)|] ^{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) }. \end{aligned}$$

(3.55)

Then, we have

$$\begin{aligned} \begin{aligned} k^{ n-2\frac{p+1}{p-1} }E_\psi (u_k^{T^L_k})&=k^{ n-2\frac{p+1}{p-1} }\sum _{z\in Z_k}E_\psi ( u_{k,z}^{T^L_k})\\&=\left( \frac{1}{2}-\frac{1}{p+1}\right) \sum _{z\in Z_k}[{\varphi }_k(z)]^{\frac{p+1}{p-1}}[k^n|T^L_k(C_z^k)|] ^{-2 \left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}}. \end{aligned} \end{aligned}$$

(3.56)

Taking into account the minimality of $T^L_k$, since $\sum _{z\in Z_k}|T^L_k(C^k_z)|=1$, we obtain

$$\begin{aligned}{}[{\varphi }_k(z)]^{\frac{p+1}{p-1}}=\mu _k[k^n|T_k^L(C^k_z)|]^{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}+1}\qquad \forall k\in {\mathbb {N}},\ \forall z\in Z_k \end{aligned}$$

(3.57)

where $\mu _k>0$ is a suitable Lagrange multiplier. It follows that

$$\begin{aligned} |T_k^L(C^k_z)|=\frac{1}{k^n}\left( \frac{[{\varphi }_k(z)]^{\frac{p+1}{p-1}}}{\mu _k}\right) ^{\frac{1}{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}+1}}\qquad \forall k\in {\mathbb {N}},\ \forall z\in Z_k \end{aligned}$$

(3.58)

which, summing up, yields

$$\begin{aligned} \mu _k^{\frac{1}{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}+1}}=\frac{1}{k^n}\sum _{z\in Z_k}[{\varphi }_k(z)]^{\frac{1}{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) +\frac{p-1}{p+1}}}. \end{aligned}$$

(3.59)

Since

$$\begin{aligned} \lim _{k\rightarrow \infty }\max \{|{\varphi }_k(z)-{\bar{{\varphi }}}|\,\ z\in Z_k\}=0 \end{aligned}$$

(3.60)

(because of (3.54)), we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{1}{k^n}\sum _{z\in Z_k}[{\varphi }_k(z)]^{\frac{1}{2\left( \frac{1}{p+1} -\frac{1}{2^*}\right) +\frac{p-1}{p+1}}}={\bar{{\varphi }}}^{\frac{1}{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) +\frac{p-1}{p+1}}}\nonumber \\ \end{aligned}$$

(3.61)

which implies

$$\begin{aligned} \lim _{k\rightarrow \infty }\mu _k={\bar{{\varphi }}}^{\frac{p+1}{p-1}}. \end{aligned}$$

(3.62)

From (3.57) we get

$$\begin{aligned} k^n|T^L_k(C^k_z)|=\left[ \frac{{\varphi }_k(z)]^{\frac{p+1}{p-1}}}{\mu _k}\right] ^{\frac{1}{2\left( \frac{1}{p+1}-\frac{1}{2^*}\right) \frac{p+1}{p-1}+1}}. \end{aligned}$$

(3.63)

Thus, we can easily obtain (3.47) from (3.60) and (3.62) and then (3.48) from (3.47) and (3.54). Finally, we can say that, for every choice of $z^k$ in ${\mathbb {Z}}_k$, the corresponding limit domain $\chi$ is a cube of ${\mathbb {R}}^n$ with sides of lenght 1. Moreover, the construction of $\chi$ shows that this cube has a vertex in the origin, so the proof is complete. $\square$

Lemma 3.8

For all $k\in {\mathbb {N}}$ and $L\in ]0,1[$, let $T^L_k \in D_L$ be a minimizing deformation as in Sect. 2. Then, for all $i\in \{1,\ldots ,n\}$, we have

$$\begin{aligned} \begin{aligned}&\lim _{k\rightarrow \infty }\sup \Bigg \{\frac{\left| f_i^{\frac{h}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z}{k}\right) -f_i^{\frac{h}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\zeta }{k}\right) \right| }{\left| P_i\circ T^L_k\left( \frac{z}{k}\right) - P_i\circ T^L_k\left( \frac{\zeta }{k}\right) \right| }\,\ h\in \{0,1,\ldots ,k\}, \\&z,\zeta \in Z_k,\ z\ne \zeta ,\ z_i=\zeta _i=h \Bigg \}=0. \end{aligned} \end{aligned}$$

(3.64)

Proof

Arguing by contradiction, assume that for some $i\in \{1,\ldots ,n\}$ there exist sequences $(h_k)_k$, $(z^k)_k$, $(\zeta ^k)_k$ such that $h_k\in \{0,1,\ldots ,k\}$, $z^k\in Z_k$, $\zeta ^k\in Z_k$, $z^k\ne \zeta ^k$, $z_i^k=\zeta ^k_i=h_k$ $\forall k\in {\mathbb {N}}$ and (up to a subsequence)

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) -f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\zeta ^k}{ k}\right) \right| }{\left| P_i\circ T^L_k\left( \frac{z^k}{k}\right) - P_i\circ T^L_k\left( \frac{\zeta ^k}{ k}\right) \right| }>0. \end{aligned}$$

(3.65)

We say that there exist two sequences $({\hat{z}}^k)_k$ and $({\hat{\zeta ^k}})_k$ in $Z_k$ such that ${\hat{z}}^k_i={\hat{\zeta }}^k_i=h_k$, $|{\hat{z}}^k-{\hat{\zeta ^k}}|=1$ $\forall k\in {\mathbb {N}}$ and

$$\begin{aligned} \liminf _{k\rightarrow \infty } \frac{\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{{\hat{z}}^k}{k}\right) -f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{{\hat{\zeta ^k}}}{ k}\right) \right| }{\left| P_i\circ T^L_k\left( \frac{ {\hat{z}}^k}{k}\right) - P_i\circ T^L_k\left( \frac{ {\hat{\zeta ^k}}}{ k}\right) \right| }>0. \end{aligned}$$

(3.66)

In fact, if $\limsup _{k\rightarrow \infty }|z^k-\zeta ^k|<2$, it is obvious because in this case $|z^k-\zeta ^k|=1$ for k large enough (so we can set ${\hat{z}}^k=z^k$ and ${\hat{\zeta }}^k=\zeta ^k$). On the contrary, if $\limsup _{k\rightarrow \infty }|z^k-\zeta ^k|\ge 2$, let us set $\nu ^k=\sum _{j=1}^n|z^k_i-\zeta _i^k|$. Then, one can choose $\nu _k+1$ points $\pi _0,\pi _1,\ldots ,\pi _{{\nu _{k}}}$ in $Z_k$ such that

$$\begin{aligned} \pi _0=z^k,\quad \pi _{\nu _k}=\zeta ^k,\quad |\pi _j-\pi _{j-1}|=1\quad \forall j\in \{1,\ldots ,\nu _k\} \end{aligned}$$

(3.67)

(notice that all the points $\frac{\pi _0}{k}$, $\frac{\pi _1}{k},\ldots ,\frac{\pi _{\nu _k}}{k}$ must belong to $F_i^{\frac{h_k}{k}}$ because of the choice of $\nu _k$).

Therefore, we obtain

$$\begin{aligned} \begin{aligned}& \left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{ z^k}{k}\right) -f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{ \zeta ^k}{ k}\right) \right| \\&\le \sum _{j=1}^{\nu _k} \left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) -f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| \\&\le \max _{1\le j\le \nu _k}\frac{\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) -f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| }{\left| P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) - P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| }\cdot \\&\quad \cdot \sum _{j=1}^{\nu _k}\left| P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) - P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| \end{aligned} \end{aligned}$$

(3.68)

where

$$\begin{aligned} \begin{aligned} \left| P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) - P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| \le \left| T^L_k\left( \frac{\pi _j}{k}\right) - T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| \le&(1+L)\left| \frac{\pi _j}{k}-\frac{\pi _{j-1}}{ k} \right| \\&\forall j\in \{1,\ldots ,\nu _k\}, \end{aligned}\nonumber \\ \end{aligned}$$

(3.69)

which implies

$$\begin{aligned} \begin{aligned} \sum _{j=1}^{\nu _k}\left| P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) - P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right|&\le (1+L)\sum _{j=1}^{\nu _k}\left| \frac{\pi _j}{k}-\frac{\pi _{j-1}}{ k} \right| \\ {}&\le (1+L)\sqrt{n-1} \left| \frac{z^k}{k}-\frac{\zeta _k}{k}\right| \\&\le \sqrt{n-1} \frac{1+L}{1-L} \left| T^L_k\left( \frac{z^k}{k}\right) - T^L_k\left( \frac{\zeta ^k}{ k}\right) \right| \end{aligned} \end{aligned}$$

(3.70)

where

(3.71)

and, by Lemma 3.1

$$\begin{aligned} \left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) -f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\zeta ^k}{ k}\right) \right| \le \frac{L}{1-L} \left| P_i\circ T^L_k\left( \frac{z^k}{k}\right) - P_i\circ T^L_k\left( \frac{\zeta ^k}{ k}\right) \right| .\nonumber \\ \end{aligned}$$

(3.72)

Therefore, (3.65) implies

$$\begin{aligned} \liminf _{k\rightarrow \infty }\max _{1\le j\le \nu _k} \frac{\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) -f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| }{\left| P_i\circ T^L_k\left( \frac{\pi _j}{k}\right) - P_i\circ T^L_k\left( \frac{\pi _{j-1}}{ k}\right) \right| }>0. \end{aligned}$$

(3.73)

So, if the maximum in (3.73) is achieved for $j= j_k$, our assertion (3.66) holds for ${\hat{z}}^k=\pi _{j_k}$ and ${\hat{\zeta }}^k=\pi _{j_k-1}$.

Now, for all $i\in \{1,\ldots ,n\}$ let us consider the vector $e^i=(e^i_1,\ldots ,e^i_n)\in {\mathbb {R}}^n$ such that $e^i_i=1$, $e^i_j=0$ for $j\ne i$, $i,j\in \{1,\ldots ,n\}$ and the function $\delta _i^k:\Omega \rightarrow {\mathbb {R}}^n$ defined by

$$\begin{aligned} \begin{aligned}&\delta ^k_i(x) =k\left[ T^L_k\left( \frac{z}{k}+\frac{e^i}{k}\right) -T^L_k\left( \frac{z}{k}\right) \right] \\&\forall x\in \Omega \ \text{ such } \text{ that } \frac{z_j}{k}\le x_j<\frac{z_j+1}{k}\quad \forall j\in \{1,\ldots ,n\}\ \text{ with } z\in Z_k. \end{aligned} \end{aligned}$$

(3.74)

Notice that the set $Z_\Omega =\cup _{k\in {\mathbb {N}}}\frac{1}{k} Z_k$ is a subset of $\overline{\Omega }$, ${\overline{Z}}_\Omega =\overline{\Omega }$ and, for all $i\in \{1,\ldots ,n\}$, the sequence of functions ${\delta _i^k}_{|_{Z_\Omega }}$, up to a subsequence, converges as $k\rightarrow \infty$ to a function $\delta _i:Z_\Omega \rightarrow {\mathbb {R}}^n$.

Taking into account Lemma 3.7, for all $x\in Z_\Omega$ we have

$$\begin{aligned} \delta _i(x)\cdot \delta _i(x)=1,\qquad \delta _i(x)\cdot \delta _j(x)=0\ \text{ for } i\ne j,\qquad \forall x\in Z_\Omega ,\ \forall i,j\in \{1,\ldots ,n\}. \end{aligned}$$

(3.75)

Moreover, we infer that

$$\begin{aligned} \liminf _{k\rightarrow \infty } \left| \delta _i^k\left( \frac{{\hat{z}}^k}{k}\right) -e^i\right| >0 \end{aligned}$$

(3.76)

because of (3.66), while

$$\begin{aligned} \lim _{k\rightarrow \infty }\delta ^k_i\left[ P_i\left( \frac{{\hat{z}}^k}{k}\right) \right] =e_i \end{aligned}$$

(3.77)

because of the conditions satisfied by $T^L_k$ on the boundary of $\Omega$.

Therefore, since

$$\begin{aligned} \lim _{k\rightarrow \infty }\left| \delta _i\left( \frac{{\hat{z}}^k}{k}\right) -\delta _i^k\left( \frac{{\hat{z}}^k}{k}\right) \right| =0 \end{aligned}$$

(3.78)

and $\delta _i\left[ P_i\left( \frac{{\hat{z}}^k}{k}\right) \right] =e^i$ $\forall k\in {\mathbb {N}}$, it follows that

$$\begin{aligned} \liminf _{k\rightarrow \infty }\left| \delta _i\left( \frac{{\hat{z}}^k}{k}\right) -\delta _i\left[ P_i\left( \frac{{\hat{z}}^k}{k}\right) \right] \right| >0. \end{aligned}$$

(3.79)

On the other hand, since ${\hat{z}}_i^k\le k$, we obtain

$$\begin{aligned} \begin{aligned} \left| \delta _i\left( \frac{{\hat{z}}^k}{k}\right) -\delta _i\left[ P_i\left( \frac{{\hat{z}}^k}{k}\right) \right] \right|&\le \sum _{j=1}^{{\hat{z}}_i^k}\left| \delta _i\left( \frac{{\hat{z}}^k-je^i}{k}\right) -\delta _i\left( \frac{{\hat{z}} ^k-(j-1)e^i}{k}\right) \right| \\&\le k\max _{1\le j\le {\hat{z}}_i^k}\left| \delta _i\left( \frac{{\hat{z}}^k-je^i}{k}\right) -\delta _i\left( \frac{{\hat{z}}^k-(j-1)e^i}{k}\right) \right| . \end{aligned} \end{aligned}$$

(3.80)

We say that the last term tends to zero as $k\rightarrow \infty$ that is, if the maximum in (3.80) is achieved for $j=j_k$, we have

$$\begin{aligned} \lim _{k\rightarrow \infty } k\left| \delta _i\left( \frac{{\hat{z}}^k-j_ke^i}{k}\right) -\delta _i\left( \frac{{\hat{z}} ^k-(j_k-1)e^i}{k}\right) \right| =0. \end{aligned}$$

(3.81)

In fact, assume that (up to a subsequence) the sequence $k\left[ \delta _i\left( \frac{{\hat{z}}^k-(j_k-1)e^i}{k}\right) -\delta _i\left( \frac{{\hat{z}}^k-j_ke^i}{k}\right) \right]$ converges as $k\rightarrow \infty$ to a vector in ${\mathbb {R}}^n$, we denote by $D_i{\hat{\delta _i}}$, and the sequences $\delta _i\left( \frac{{\hat{z}}^k-(j_k-1)e^i}{k}\right)$ converge to some vectors ${\hat{\delta _i}}$ in ${\mathbb {R}}^n$. Then, we have that ${\hat{\delta }_i\cdot {\hat{\delta }_j}}=0$ for $i\ne j$ and ${\hat{\delta }_i\cdot {\hat{\delta }_i}}=1$ for $i,j\in \{1,\ldots ,n\}$ (because of Lemma 3.7). Therefore, in order to prove that $D_i{\hat{\delta }_i}=0$, it suffices to prove that $D_i{\hat{\delta }_i\cdot {\hat{\delta }_j}}=0$ $\forall j\in \{1,\ldots ,n\}$.

First, notice that $D_i{\hat{\delta }_i\cdot {\hat{\delta }_i}}=0$. In fact, we have

$$\begin{aligned} D_i{\hat{\delta _i}\cdot {\hat{\delta _i}}}=\frac{1}{2}\lim _{k\rightarrow \infty } k\left[ \delta _i\cdot \delta _i\left( \frac{{\hat{z}}^k-(j_k-1)e^i}{k}\right) -\delta _i\cdot \delta _i\left( \frac{{\hat{z}}^k-j_ke^i}{k}\right) \right] , \end{aligned}$$

(3.82)

where the limit is equal to zero because

$$\begin{aligned} \delta _i\cdot \delta _i\left( \frac{{\hat{z}}^k-j_ke^i}{k}\right) =\delta _i\cdot \delta _i\left( \frac{{\hat{z}}^k-(j_k-1)e^i}{k}\right) =1 \qquad \forall k\in {\mathbb {N}}. \end{aligned}$$

(3.83)

In order to prove that $D_i{{\hat{\delta }_i}\cdot {\hat{\delta }_j}}=0$ for $j\ne i$, notice that, since $\delta _i\cdot \delta _j\equiv 0$ in $Z_\Omega$, we have

$$\begin{aligned} 0=\lim _{k\rightarrow \infty }k\left[ \delta _i\cdot \delta _j\left( \frac{{\hat{z}}^k-(j_k-1)e^i}{k}\right) -\delta _i\cdot \delta _j\left( \frac{{\hat{z}}^k-j_ke^i}{k}\right) \right] =D_i{\hat{\delta }_i\cdot {\hat{\delta }_j}} +{\hat{\delta }_i\cdot } D_i{\hat{\delta }_j}.\nonumber \\ \end{aligned}$$

(3.84)

Moreover, we have

$$\begin{aligned} \begin{aligned} D_i{\hat{\delta }}_j=D_j{\hat{\delta _i}}=\lim _{k\rightarrow \infty } k^2\Bigg [&T_k^L({\hat{z}}^k-(j_k-1)e^i+e^j)-T^L_k\left( \frac{{\hat{z}}^k-(j_k-1)e^i}{k}\right) \\&- T^L_k\left( \frac{{\hat{z}}^k-j_ke^i+e^j}{k}\right) + T^L_k\left( \frac{{\hat{z}}^k-j_ke^i}{k}\right) \Bigg ] \end{aligned} \end{aligned}$$

(3.85)

and, as a consequence,

$$\begin{aligned} {\hat{\delta _i\cdot }} D_i{\hat{\delta }}_j={\hat{\delta _i\cdot }} D_j{\hat{\delta _i}}= \frac{1}{2} \lim _{k\rightarrow \infty } k\Bigg [ \delta _i\cdot \delta _i \left( \frac{{\hat{z}}^k-j_k e^i+e^j}{k}\right) - \delta _i\cdot \delta _i\left( \frac{{\hat{z}}^k-j_ke^i }{k}\right) \Bigg ]=0 \nonumber \\ \end{aligned}$$

(3.86)

because $\delta _i\cdot \delta _i\equiv 1$ in $Z_\Omega$. Therefore, from (3.84) we obtain $D_i{\hat{\delta }_i\cdot \hat{\delta }_j}=0$ also for $i\ne j$, so $D_i{\hat{\delta }_i}=0$.

Then, from (3.80) we infer that

$$\begin{aligned} \lim _{k\rightarrow \infty }\left| \delta _i\left( \frac{{\hat{z}}^k}{k}\right) -\delta _i\left[ P_i\left( \frac{{\hat{z}}^k}{k}\right) \right] \right| =0 \end{aligned}$$

(3.87)

in contradiction with (3.79).

Thus, we can conclude that (3.65) cannot hold and (3.79) is true. So the proof is complete. $\square$

Indeed, the minimality of $T^L_k$ allows us to prove a stronger result, stated in the following corollary.

Corollary 3.9

Under the same assumptions of Lemma 3.8, for all $i\in \{1,\ldots ,n\}$ we have

$$\begin{aligned} \lim _{k\rightarrow \infty } \sup \left\{ \frac{|f_i^{\frac{h}{k}, T^L_k}(x)-f_i^{\frac{h}{k}, T^L_k}(y)|}{|x-y|}\,\ h\in \{0,1,\ldots ,k\},\ x,y\in F^0_i,\ x\ne y\right\} =0. \qquad \end{aligned}$$

(3.88)

Proof

Since, under our assumptions on the values of $T^L_k$ on $\partial \Omega$, $P_i\circ T^L_k$ is a one-to-one map between $F^{\frac{h}{k}}_0$ and $F^0_i$, (3.88) is equivalent to

$$\begin{aligned} \begin{aligned}&\lim _{k\rightarrow \infty } \sup \Bigg \{\frac{|f_i^{\frac{h}{k}, T^L_k}\circ P_i\circ T^L_k(x)-f_i^{\frac{h}{k}, T^L_k}\circ P_i\circ T^L_k(y)|}{ |P_i\circ T^L_k(x)- P_i\circ T^L_k(y)|}\,\ h\in \{0,1,\ldots ,k\},\\&x,y\in F^{\frac{h}{k}}_i,\ x\ne y\Bigg \}=0. \end{aligned} \end{aligned}$$

(3.89)

Arguing by contradiction, assume that there exist sequences $(h_k)_k$, $(x^k)_k$, $(y^k)_k$ such that $h_k\in \{0,1,\ldots ,k\}$, $x^k$ and $y^k$ belong to $F_i^{\frac{h}{k}}$, $x^k\ne y^k$ $\forall k\in {\mathbb {N}}$ and (up to a subsequence)

$$\begin{aligned} \lim _{k\rightarrow \infty } \frac{|f_i^{\frac{h_k}{k}, T^L_k}\circ P_i\circ T^L_k(x^k)-f_i^{\frac{h_k}{k}, T^L_k}\circ P_i\circ T^L_k(y^k)|}{ |P_i\circ T^L_k(x^k)- P_i\circ T^L_k(y^k)|}>0. \end{aligned}$$

(3.90)

Notice that the interfaces between the domains $kT^L_k(C^k_z)$, $z\in Z_k$, tend to be flat because of the minimality of the admissible deformation $T^L_k$ and, as follows from Lemmas 3.7 and 3.8, up to translations these domains tend to $\Omega$ that is, for every choice of $z^k$ in $Z_k$, the domain $k\left[ T^L_k(C^k_z)-T^L_k\left( \frac{z^k}{k}\right) \right]$ tends to $C^1_0=\Omega$.

Therefore, (3.90) is possible only if $\lim _{k\rightarrow \infty }k|x^k-y^k|=\infty$ (otherwise, up to a subsequence, the segment $\{x^k+t(y^k-x^k)\,\ t\in [0,1]\}$ meets only a finite number of subdomains ${\overline{C}}^k_z$ with $z\in Z_k$). In this case, if $x^k\in C^k_{z^k}$ and $y^k\in C^k_{\zeta ^k}$ for suitable $z^k$ and $\zeta ^k$ in $Z_k$, we have

$$\begin{aligned} |kx^k-z^k|\le \sqrt{n},\quad |ky^k-\zeta ^k|\le \sqrt{n}\qquad \forall k\in {\mathbb {N}},\qquad \lim _{k\rightarrow \infty }|z^k-\zeta ^k|=\infty \end{aligned}$$

(3.91)

and

$$\begin{aligned} \begin{aligned}&\Big |f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(x^k)- f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(y^k)\Big | \\&\quad \le \left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(x^k)- f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right| \\&\quad \quad + \left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) - f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) \right| \\&\quad \quad +\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) - f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(y^k)\right| \end{aligned} \end{aligned}$$

(3.92)

where

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) - f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) \right| }{\left| P_i\circ T^L_k\left( \frac{z^k}{k}\right) - P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) \right| } =0 \end{aligned}$$

(3.93)

(as follows from Lemma 3.8) and

$$\begin{aligned}{} & {} \lim _{k\rightarrow \infty }\frac{\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( x^k\right) - f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right| }{\left| P_i\circ T^L_k\left( x^k\right) - P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right| } =0, \end{aligned}$$

(3.94)

$$\begin{aligned}{} & {} \lim _{k\rightarrow \infty }\frac{\left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( y^k\right) - f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) \right| }{\left| P_i\circ T^L_k\left( y^k\right) - P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) \right| } =0 \end{aligned}$$

(3.95)

because the segments $\left\{ x^k+t\left( \frac{z^k}{k}-x^k\right) \,\ t\in [0,1]\right\}$ and $\left\{ y^k+t\left( \frac{\zeta ^k}{k}-y^k\right) \,\ t\in [0,1]\right\}$ are respectively enclosed in the subdomains ${\overline{C}}^k_{z^k}$ and ${\overline{C}}^k_{\zeta ^k}$.

Moreover, for k large enough, we have

$$\begin{aligned} \begin{aligned}&\left| P_i\circ T^L_k(x^k)-P_i\circ T^L_k\left( \frac{z^k}{k}\right) \right| \le \left| T^L_k(x^k)- T^L_k\left( \frac{z^k}{k}\right) \right| \le (1+L)\left( x^k-\frac{z^k}{k}\right) \\&\le (1+L)|x^k-y^k|\le \frac{1+L}{1-L}\ |T^L_k(x^k)-T^L_k(y^k)| \end{aligned} \end{aligned}$$

(3.96)

and, analogously,

$$\begin{aligned} \left| P_i\circ T^L_k(y^k)-P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) \right| \le (1+L)|x^k-y^k|\le \frac{1+L}{1-L}\ |T^L_k(x^k)-T^L_k(y^k)| \nonumber \\ \end{aligned}$$

(3.97)

with

$$\begin{aligned}{} & {} |T^L_k(x^k)-T^L_k(y^k)|\le |P_i\circ T^L_k(x^k)-P_i\circ T^L_k(y^k)|\nonumber \\{} & {} \quad + \left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(x^k)-f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(y^k)\right| \end{aligned}$$

(3.98)

where

$$\begin{aligned} \left| f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(x^k)-f_i^{\frac{h_k}{k},T^L_k}\circ P_i\circ T^L_k(y^k)\right| \le \frac{L}{1-L}\ |P_i\circ T^L_k(x^k)-P_i\circ T^L_k(y^k)| \nonumber \\ \end{aligned}$$

(3.99)

because of Lemma 3.1.

In a similar way we obtain

$$\begin{aligned} \begin{aligned}&\Big | P_i\circ T^L_k\left( \frac{z^k}{k}\right) -P_i \circ T^L_k\left( \frac{\zeta ^k}{k}\right) \Big | \le \left| P_i\circ T^L_k\left( x^k\right) -P_i\circ T^L_k\left( y^k\right) \right| \\&\quad +\left| P_i\circ T^L_k\left( \frac{z^k}{k}\right) -P_i\circ T^L_k\left( x^k\right) \right| +\left| P_i\circ T^L_k\left( \frac{\zeta ^k}{k}\right) -P_i\circ T^L_k\left( y^k\right) \right| . \end{aligned} \end{aligned}$$

(3.100)

Therefore, it follows easily that (3.90) cannot be true.

Thus, we have a contradiction, so (3.88) holds and the proof is complete. $\square$

Lemma 3.10

Under the same assumptions as in Lemma 3.8, for all $i\in \{1,\ldots ,n\}$ we have also

$$\begin{aligned} \lim _{k\rightarrow \infty }\sup \big \{|k\,[f^{\frac{h}{k},T^L_k}_i(x)-f^{\frac{h-1}{k},T^L_k}_i(x)]-1|\,\ h\in \{1,\ldots ,k\},\ x\in F_i^0\big \}=0. \end{aligned}$$

(3.101)

Proof

Arguing by contradiction, assume that for all $k\in {\mathbb {N}}$ there exist $h_k\in \{1,\ldots ,k\}$ and $x^k\in F^0_i$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty } k \left[ f_i^{\frac{h_k}{k},T^L_k}(x^k)- f_i^{\frac{h_k-1}{k},T^L_k}(x^k) \right] \ne 1 \end{aligned}$$

(3.102)

for some $i\in \{1,\ldots ,n\}$.

For all $k\in {\mathbb {N}}$, let us choose $y^k\in \Omega$ and $z^k\in Z_k$ such that

$$\begin{aligned} y^k\in C^k_{z^k},\ P_i(y^k)=x^k \text{ and } f_i^{\frac{h_k-1}{k},T^L_k}(x^k)\le y^k_i\le f_i^{\frac{h_k}{k},T^L_k}(x^k)\qquad \forall k\in {\mathbb {N}}. \end{aligned}$$

(3.103)

Therefore, we have

$$\begin{aligned} T^L_k({\overline{C}}^k_{z^k})=\left\{ y\in \Omega \,\ f_i^{\frac{h_k-1}{k},T^L_k}\circ P_i(y)\le y_i\le f_i^{\frac{h_k}{k},T^L_k}\circ P_i(y) \text{ for } i=1,\ldots ,n\right\} \quad \forall k\in {\mathbb {N}}. \qquad \nonumber \\ \end{aligned}$$

(3.104)

Taking into account Lemmas 3.7, 3.8 and Corollary 3.9, the domain $k\left[ T^k_L({\overline{C}}^k_{z^k})-T^L_k\left( \frac{z^k}{k}\right) \right]$ tends to ${\overline{C}}^1_0=\overline{\Omega }$ as $k\rightarrow \infty$.

On the other hand, this convergence is not possible if (3.102) holds for some $i\in \{1,\ldots ,n\}$.

Thus, we have a contradiction, (3.102) cannot hold for any $i\in \{1,\ldots ,n\}$ and (3.101) is true. So the proof is complete. $\square$

Now, for all $i\in \{1,\ldots ,n\}$, $t\in [0,1]$ and $k\in {\mathbb {N}}$, let us consider the function ${\tilde{f}}_i^{t,k,L}:F^0_i\rightarrow [0,1]$ defined by

$$\begin{aligned} \begin{aligned}&{\tilde{f}}_i^{t,k,L}(x)= f_i^{\frac{h-1}{k},T^L_k}(x)+(kt-h+1)[f_i^{\frac{h}{k},T^L_k}(x) -f_i^{\frac{h-1}{k},T^L_k}(x)] \\&\forall x\in F^0_i,\ \forall t\in \left[ \frac{h-1}{k},\frac{h}{k}\right] ,\ h\in \{1,\ldots ,k\}. \end{aligned} \end{aligned}$$

(3.105)

Proposition 3.11

For all $i\in \{1,\ldots ,n\}$, $t\in [0,1]$ and $k\in {\mathbb {N}}$, the functions ${\tilde{f}}_i^{t,k,L}$ defined by (3.105) have the following properties:

$$\begin{aligned} \lim _{k\rightarrow \infty }\sup \left\{ \textrm{Lip}({\tilde{f}}_i^{t,k,L})\,\ i\in \{1,\ldots ,n\},\ t\in [0,1]\right\} =0 \end{aligned}$$

(3.106)

$$\begin{aligned} \begin{aligned}&\lim _{k\rightarrow \infty }\sup \Bigg \{\frac{1}{|t_1-t_2|}\,|{\tilde{f}}_i^{t_1,k,L}(x)- {\tilde{f}}_i^{t_2,k,L}(x)-t_1+t_2|\,\\ {}&i\in \{1,\ldots ,n\},\ t_1,t_2\in [0,1], \ t_1\ne t_2,\ x\in F^0_i\Bigg \}=0. \end{aligned} \end{aligned}$$

(3.107)

Proof

Taking into account the definition of ${\tilde{f}}_i^{t,k,L}$, properties (3.106) and (3.107) follow by direct computation respectively from Corollary 3.9 and Lemma 3.10. $\square$

Lemma 3.12

Let ${\tilde{f}}_i^{t,k,L}$ be the functions defined in (3.105). Then, for all $x\in \overline{\Omega }$ there exists $y\in \overline{\Omega }$ such that ${\tilde{f}}_i^{x_i,k,L}\circ P_i(y)=y_i$ $\forall i\in \{1,\ldots ,n\}$ (that is, y belongs to the graph of ${\tilde{f}}_i^{x_i,k,L}$ for $i=1,\ldots ,n$).

Proof

From Proposition 2.3 and (3.105) we infer that ${\tilde{f}}_i^{t,k,L}\circ P_i(y)$ is strictly increasing with respect to t in the interval [0, 1] for all $i\in \{1,\ldots ,n\}$, $k\in {\mathbb {N}}$, $y\in \overline{\Omega }$. Moreover, we have

$$\begin{aligned} 0={\tilde{f}}_i^{0,k,L}\circ P_i(y)\le y_i\le {\tilde{f}}_i^{1,k,L}\circ P_i(y)=1 \end{aligned}$$

(3.108)

so for all $y\in \overline{\Omega }$ there exists a unique $t_i(y)\in [0,1]$ such that ${\tilde{f}}_i^{t_i(y),k,L}\circ P_i(y)=y_i$. Let us set $t(y)=(t_1(y),\ldots ,t_n(y))$. Then, the function t(y), defined for all $y\in \overline{\Omega }$, is continuous in $\overline{\Omega }$ and satisfies

$$\begin{aligned} t(F^0_i)\subseteq F^0_i\quad \text{ and } \quad t(F^1_i)\subseteq F^1_i\qquad \forall i\in \{1,\ldots ,n\}. \end{aligned}$$

(3.109)

Therefore, from [33] we infer that for all $x\in \overline{\Omega }$ there exists at least one $y\in \overline{\Omega }$ such that $t(y)=x$, that is $t_i(y)=x_i$ for all $i\in \{1,\ldots ,n\}$.

Thus, since $t_i(y)=x_i$ is equivalent to ${\tilde{f}}_i^{x_i,k,L}\circ P_i(y)=y_i$, the proof is complete. $\square$

Lemma 3.13

Let ${\tilde{f}}_i^{t,k,L}$ be the functions defined in (3.105). Then, there exists ${\tilde{k}}_1\in {\mathbb {N}}$ such that for all $k\ge {\tilde{k}}_1$ the following property holds: for all $x\in \overline{\Omega }$ there exists a unique $y\in \overline{\Omega }$ such that

$$\begin{aligned} {\tilde{f}}_i^{x_i,k,L}\circ P_i(y)=y_i\qquad \forall i\in \{1,\ldots ,n\}. \end{aligned}$$

(3.110)

Proof

In Lemma 3.12 we proved that for all $k\in {\mathbb {N}}$ and for all $x\in \overline{\Omega }$ there exists at least one $y\in \overline{\Omega }$ satisfying (3.110). Now, we have to prove that for k large enough such a y is unique.

For all ${{\mathcal {L}}}\ge 0$ let us set $C_i({{\mathcal {L}}})=\{x\in {\mathbb {R}}^n\,\ |x_i|\le {{\mathcal {L}}}|P_i(x)|\}$ and notice that the graph of ${\tilde{f}}_i^{x_i,k,L}$ is enclosed in $y+C_i(\textrm{Lip}({\tilde{f}}_i^{x_i,k,L}))$.

One can verify by direct computation that $\cap _{1\le i\le n}C_i({{\mathcal {L}}}_i)=\{0\}$ when ${{\mathcal {L}}}_i<(n-1)^{-\frac{1}{2}}$ $\forall i\in \{1,\ldots ,n\}$.

Therefore, if $\textrm{Lip}({\tilde{f}}_i^{x_i,k,L})<(n-1)^{-\frac{1}{2}}$ $\forall i\in \{1,\ldots ,n\}$, y is the unique point in ${\mathbb {R}}^n$ satisfying (3.110). On the other hand, taking into account (3.106) of Proposition 3.11, we infer that there exists ${\tilde{k}}_1\in {\mathbb {N}}$ such that

$$\begin{aligned} \textrm{Lip}({\tilde{f}}_i^{t,k,L})<(n-1)^{-\frac{1}{2}}\qquad \forall k\ge {\tilde{k}}_1,\ \forall i\in \{1,\ldots ,n\},\ \forall t\in [0,1]. \end{aligned}$$

(3.111)

Thus, the assertion of Lemma 3.13 holds for such a ${\tilde{k}}_1$, so the proof is complete. $\square$

Definition 3.14

Taking into account Lemma 3.13, for all $k\ge {\tilde{k}}_1$ we can define a function ${\widetilde{T}}_k^L:\overline{\Omega }\rightarrow \overline{\Omega }$ in the following way. For all $x\in \overline{\Omega }$ we set ${\widetilde{T}}^L_k(x)=y$ where y is the unique point in $\overline{\Omega }$ satisfying (3.110), given by Lemma 3.13

Taking into account the properties of the function ${\tilde{f}}_i^{t,k,L}$ defined by (3.105) one can verify by standard arguments that ${\widetilde{T}}_k^L$ is a one-to-one continuous function and that $({\widetilde{T}}_k^L)^{-1}$ is continuous too. Moreover, for all $i\in \{1,\ldots ,n\}$ and $t\in [0,1]$, ${\widetilde{T}}^L_k(F^t_i)$ is the graph of the function ${\tilde{f}}_i^{t,k,L}$ (that is $f_i^{t,{\widetilde{T}}^L_k}={\tilde{f}}_i^{t,k,L}$), ${\widetilde{T}}^L_k(F^t_i)=F^t_i$ for $t\in \{0,1\}$ and

$$\begin{aligned} {\widetilde{T}}_k^L(C_z^k)=T^L_k(C_z^k)\qquad \forall k\ge {\tilde{k}}_1,\ \forall z\in Z_k. \end{aligned}$$

(3.112)

Proposition 3.15

For all $k\ge {\tilde{k}}_1$ and $L\in ]0,1[$, let ${\widetilde{T}}_k^L$ be the function introduced in Definition 3.14. Then, there exists ${\tilde{k}}_2\in {\mathbb {N}}$ such that ${\widetilde{T}}^L_k\in D_L$ $\forall k\ge {\tilde{k}}_2$. Moreover, there exists a sequence of positive numbers $(L_k)_k$ such that $\lim _{k\rightarrow \infty } L_k=0$ and ${{\widetilde{T}}}_k^L\subseteq D_{L_k}$ $\forall k\ge {\tilde{k}}_2$.

Proof

From Proposition 3.11 we infer that there exists a sequence of positive numbers $(\Lambda _k)_k$ such that $\lim _{k\rightarrow \infty }\Lambda _k=0$ and

$$\begin{aligned}{} & {} \textrm{Lip}({\tilde{f}}_i^{t,k,L})\le \Lambda _k\qquad \forall k\in {\mathbb {N}},\ \forall i\in \{1,\ldots ,n\},\ \forall t\in [0,1], \end{aligned}$$

(3.113)

$$\begin{aligned}{} & {} |{\tilde{f}}_i^{t_1,k,L}(x)-{\tilde{f}}_i^{t_2,k,L}(x)-t_1+t_2|\le \Lambda _k|t_1-t_2|\qquad \forall k\in {\mathbb {N}},\ \forall i\in \{1,\ldots ,n\},\ \forall x\in F^0_i. \nonumber \\ \end{aligned}$$

(3.114)

Since $\Lambda _k\rightarrow 0$, we can choose $k_n\in {\mathbb {N}}$ such that $\Lambda _k<\frac{1}{n}$ $\forall k\ge k_n$. Hence, taking into account that ${\tilde{f}}_i^{t,k,L}=f_i^{t,{{\widetilde{T}}}_k^L}$, from Lemma 3.2 we infer that, for all $k\ge k_n$, ${{\widetilde{T}}}_k^L\in D_{L_k}$ where $L_k=\frac{(n+1)\sqrt{n}\, \Lambda _k}{1-n\Lambda _k}$. Since $L_k\rightarrow 0$ as $k\rightarrow \infty$, we can choose ${\tilde{k}}_2$ such that $L_k\le L$ $\forall k\ge {\tilde{k}}_2$. So the proof is complete. $\square$

Now, we can show that the function $u_k=u_k^{T^L_k}$ is a solution of problem (1.1) for k large enough and satisfies all the assertions of Proposition 2.2 (in a similar way one can argue for the function $v_k$).

Notice that $u_k^{T_k^L}=u_k^{{{\widetilde{T}}}_k^L}$, where ${{\widetilde{T}}}_k^L\in D_L$ is the function introduced in Definition 3.14, because $T^L_k(C_z^k)={{\widetilde{T}}}_k^L(C_z^L)$ $\forall z\in Z_k$.

First, we prove that $u_k^{{{\widetilde{T}}}_k^L}$ is a solution of the Dirichlet problem in every subdomain ${{\widetilde{T}}}_k^L(C_z^k)$ for all $z\in Z_k$ and then we show that it satisfies a suitable stationarity condition which allows us to prove that, indeed, it is a solution of the Dirichlet problem (1.1) in the domain $\Omega$.

Lemma 3.16

There exists $k_1(L)\in {\mathbb {N}}$ such that, for all $k\ge k_1(L)$ and $z\in Z_k$, the function $u_{k,z}^{{{\widetilde{T}}}_k^L}$ is a solution of the Dirichlet problem

$$\begin{aligned} -\Delta u=|u|^{p-1}u+\psi \quad \text{ in } \ {{\widetilde{T}}}_k^L(C_z^k),\qquad u=0\quad \text{ on } \ \partial {{\widetilde{T}}}_k^L(C_z^k). \end{aligned}$$

(3.115)

Proof

For all $k\in {\mathbb {N}}$ and $z\in Z_k$, let us consider the function $G_{k,z}^{{{\widetilde{T}}}_k^L}:{{\widetilde{T}}}_k^L(C_z^k)\times {\mathbb {R}}\rightarrow {\mathbb {R}}$ such that $G_{k,z}^{{{\widetilde{T}}}_k^L}(x,\cdot )\in {{\mathcal {C}}}^2({\mathbb {R}})$ $\forall x\in {{\widetilde{T}}}_k^L(C_z^k)$ and

$$\begin{aligned} G_{k,z}^{{{\widetilde{T}}}_k^L}(x,t)= & {} \frac{|t|^{p+1}}{p+1}+\psi (x)t\quad \ \,\quad \text{ if } \sigma (z)[t-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}(x)]\ge 0,\end{aligned}$$

(3.116)

$$\begin{aligned} \frac{\partial ^2 G_{k,z}^{{{\widetilde{T}}}_k^L}(x,t)}{\partial t^2}= & {} \frac{\partial ^2 G_{k,z}^{{{\widetilde{T}}}_k^L}}{\partial t^2} (x,{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}(x))\quad \text{ if } \sigma (z)[t-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}(x)]\le 0, \end{aligned}$$

(3.117)

where ${\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}$ is the function given by Lemma 2.4 which, for k large enough, is a solution of problem (3.115) because it is a local minimum of the functional $E_\psi$ in $H^1_0({{\widetilde{T}}}_k^L(C_z^k))$.

Moreover, let us set $g_{k,z}^{{{\widetilde{T}}}_k^L}(x,t)=\frac{\partial G_{k,z}^{{{\widetilde{T}}}_k^L}}{\partial t}(x,t).$ Then, let us consider the functional $E_{k,z,{{\widetilde{T}}}_k^L}:H^1_0({{\widetilde{T}}}_k^L(C_z^k))\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} E_{k,z,{{\widetilde{T}}}_k^L}(u)=\frac{1}{2}\int _{{{\widetilde{T}}}_k^L(C_z^k)}|{\nabla }u|^2dx- \int _{{{\widetilde{T}}}_k^L(C_z^k)}G_{k,z}^{{{\widetilde{T}}}_k^L}(x,u)\, dx. \end{aligned}$$

(3.118)

Since $p>1$, for k large enough one can verify that for all $u\not \equiv {\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}$ there exists $t_u>0$ such that

$$\begin{aligned} E'_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t_u(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big ) [u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}]=0 \end{aligned}$$

(3.119)

if and only if $\sigma (z)[u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}(x)]\vee 0\not \equiv 0$; in this case such a $t_u$ is unique; in the other case we have

$$\begin{aligned} E'_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big ) [u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}]>0\quad \forall t>0 \end{aligned}$$

(3.120)

and

$$\begin{aligned} \lim _{t\rightarrow \infty } E_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big )=\infty . \end{aligned}$$

(3.121)

When $\sigma (z)[u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}(x)]\vee 0\not \equiv 0$, one can verify by direct computation that $E'_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}) \big )[u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}]$ is positive for $t\in ]0,t_u[$ and negative for $t>t_u$, so

$$\begin{aligned} E_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t_u(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big ) =\max \{ E_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big )\,\ t>0\}. \end{aligned}$$

(3.122)

Moreover, one can verify that

$$\begin{aligned} E''_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t_u(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big ) [u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L},u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}]<0. \end{aligned}$$

(3.123)

Assume, for example, that $\sigma (z)=1$ (in a similar way one can argue when $\sigma (z)=-1$). In this case, we have

$$\begin{aligned} \begin{aligned}&E_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big )\\&\quad = E_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}) \vee 0]\big ) +t E'_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}\big )[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0] \\&+\frac{t^2}{2}\int _{{{\widetilde{T}}}_k^L(C_z^k) }|{\nabla }[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0]|^2dx \\&-\frac{t^2}{2}\, p\int _{{{\widetilde{T}}}_k^L(C_z^k) }|{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}|^{p-1}[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0]^2dx\qquad \forall t>0, \end{aligned} \end{aligned}$$

(3.124)

where

$$\begin{aligned} E'_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}\big )[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0]=0 \end{aligned}$$

(3.125)

because ${\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}$ is a solution of problem (3.115).

Notice that

$$\begin{aligned} \begin{aligned}&\int _{{{\widetilde{T}}}_k^L(C_z^k) }|{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}|^{p-1}[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0]^2dx\\&\quad \le \left( \int _{{{\widetilde{T}}}_k^L(C_z^k) }|{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}|^{p+1}dx\right) ^{\frac{p-1}{p+1}} \left( \int _{{{\widetilde{T}}}_k^L(C_z^k) } |(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0|^{p+1}dx\right) ^{\frac{2}{p+1}} \end{aligned} \end{aligned}$$

(3.126)

where, as follows from Lemma 2.4,

$$\begin{aligned} \lim _{k\rightarrow \infty }\max \left\{ \int _{{{\widetilde{T}}}_k^L(C_z^k) }|{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}|^{p+1}dx\,\ z\in Z_k\right\} =0. \end{aligned}$$

(3.127)

Moreover, we have

$$\begin{aligned} \int _{{{\widetilde{T}}}_k^L(C_z^k) }|{\nabla }[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0]|^2dx\ge \lambda _k \left( \int _{{{\widetilde{T}}}_k^L(C_z^k) }|(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\wedge 0 |^{p+1}dx\right) ^{\frac{2}{p+1}}\quad \forall k\in {\mathbb {N}},\ \forall z\in Z_k\nonumber \\ \end{aligned}$$

(3.128)

where, for all $k\in {\mathbb {N}}$,

$$\begin{aligned} \lambda _k=\min \left\{ \int _{{{\widetilde{T}}}_k^L(C_z^k) }|{\nabla }v|^2dx\,\ z\in Z_k,\ v\in H^1_0\big ({{\widetilde{T}}}_k^L(C_z^k)\big ),\ \int _{{{\widetilde{T}}}_k^L(C_z^k) }| v|^{p+1}dx=1\right\} .\nonumber \\ \end{aligned}$$

(3.129)

We say that $\lim _{k\rightarrow \infty }\lambda _k=\infty$. In fact, otherwise, there exist suitable sequences $(z^k)_k$ in ${\mathbb {R}}^n$, and $(v_k)_k$ in $H^1_0(\Omega )$ such that $z^k\in Z_k$, $v_k\equiv 0$ in $\Omega {\setminus } C_{z^k}^k$, $\int _\Omega |v_k|^{p+1}dx=1$ $\forall k\in {\mathbb {N}}$ and (up to a subsequence) $\lim _{k\rightarrow \infty }\int _\Omega |{\nabla }v_k|^2dx<\infty$.

As a consequence, since $p<\frac{n+2}{n-2}$ when $n\ge 3$, there exists ${\bar{v}}\in H^1_0(\Omega )$ such that (up to a subsequence) $v_k\rightarrow {\bar{v}}$ as $k\rightarrow \infty$ weakly in $H^1_0(\Omega )$, in $L^{p+1}(\Omega )$ and almost everywhere in $\Omega$.

Taking into account that $\lim _{k\rightarrow \infty }\textrm{meas}{{\widetilde{T}}}_k^L(C_z^k)=0$, the almost everywhere convegence implies that ${\bar{v}}\equiv 0$ in $\Omega$, in contradiction with the convergence in $L^{p+1}(\Omega )$ because $\int _{{{\widetilde{T}}}_k^L(C_z^k) }|v_k|^{p+1}$ $dx=1$ $\forall k\in {\mathbb {N}}$. Thus, we can conclude that $\lim _{k\rightarrow \infty }\lambda _k=\infty$.

It follows that, for k large enough,

$$\begin{aligned} E_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t[(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\vee 0]\big )\le E_{k,z,{{\widetilde{T}}}_k^L}\big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}+t(u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big )\qquad \forall t>0,\ \forall u\in H^1_0(C^k_z).\nonumber \\ \end{aligned}$$

(3.130)

As a consequence, if we set

$$\begin{aligned} \Gamma =\{u\in H^1_0({{\widetilde{T}}}_k^L(C_z^k))\,\ u\not \equiv {\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L},\ E'_{k,z,{{\widetilde{T}}}_k^L}(u)[u-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}]=0\}, \end{aligned}$$

(3.131)

we have $u_{k,z}^{{{\widetilde{T}}}_k^L}\in \Gamma$ and

$$\begin{aligned} E_\psi ( u_{k,z}^{{{\widetilde{T}}}_k^L} )=E_{k,z,{{\widetilde{T}}}_k^L}( u_{k,z}^{{{\widetilde{T}}}_k^L})=\min _\Gamma E_{k,z,{{\widetilde{T}}}_k^L}. \end{aligned}$$

(3.132)

Therefore, there exists a Lagrange multiplier $\mu \in {\mathbb {R}}$ such that

$$\begin{aligned} E'_{k,z,{{\widetilde{T}}}_k^L}( u_{k,z}^{{{\widetilde{T}}}_k^L})[v]=\mu \left\{ E''_{k,z,{{\widetilde{T}}}_k^L}( u_{k,z}^{{{\widetilde{T}}}_k^L})[u_{k,z}^{{{\widetilde{T}}}_k^L}-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L},v] +E'_{k,z,{{\widetilde{T}}}_k^L}( u_{k,z}^{{{\widetilde{T}}}_k^L})[v]\right\} \qquad \forall v\in H^1_0(C_z^k).\nonumber \\ \end{aligned}$$

(3.133)

In particular, if we choose $v= u_{k,z}^{{{\widetilde{T}}}_k^L}-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}$, we obtain $\mu =0$ because

$$\begin{aligned} E'_{k,z,{{\widetilde{T}}}_k^L}\left( u_{k,z}^{{{\widetilde{T}}}_k^L}\right) \left[ u_{k,z}^{{{\widetilde{T}}}_k^L}-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}\right] =0 \end{aligned}$$

(3.134)

while

$$\begin{aligned} E''_{k,z,{{\widetilde{T}}}_k^L}( u_{k,z}^{{{\widetilde{T}}}_k^L})\left[ u_{k,z}^{{{\widetilde{T}}}_k^L}-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L},u_{k,z}^{{{\widetilde{T}}}_k^L}-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}\right] \ne 0. \end{aligned}$$

(3.135)

Thus, $u_{k,z}^{{{\widetilde{T}}}_k^L}$ is a weak solution of the Dirichlet problem

$$\begin{aligned} -\Delta u_{k,z}^{{{\widetilde{T}}}_k^L}=g\left( x, u_{k,z}^{{{\widetilde{T}}}_k^L}\right) \quad \text{ in } {{\widetilde{T}}}_k^L(C^k_z),\qquad u=0\quad \text{ on } \partial {{\widetilde{T}}}_k^L(C^k_z). \end{aligned}$$

(3.136)

On the other hand, since $u_{k,z}^{{{\widetilde{T}}}_k^L}\ge {\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}$ in ${{\widetilde{T}}}_k^L(C^k_z)$, we have

$$\begin{aligned} g\big (x, u_{k,z}^{{{\widetilde{T}}}_k^L}(x)\big )=| u_{k,z}^{{{\widetilde{T}}}_k^L}(x)|^{p-1}u_{k,z}^{{{\widetilde{T}}}_k^L}(x) +\psi (x)\qquad \forall x\in {{\widetilde{T}}}_k^L(C^k_z). \end{aligned}$$

(3.137)

So $u_{k,z}^{{{\widetilde{T}}}_k^L}$ is a solution of problem (3.115) and the proof is complete. $\square$

Proposition 3.17

Under the assumptions of Proposition 2.2, there exists ${\bar{k}}\in {\mathbb {N}}$ such that the function $u^{{{\widetilde{T}}}_k^L}_k=\sum _{z\in Z_k}u_{k,z}^{{{\widetilde{T}}}_k^L}$ is a solution of problem (1.1) for all $k\ge {\bar{k}}$.

Proof

From Lemma 3.16 we infer that, for a suitable $k_1(L)\in {\mathbb {N}}$, $E_\psi '\left( u_{k}^{{{\widetilde{T}}}_k^L}\right) [v]=0$ for all $v\in H^1_0({{\widetilde{T}}}_k^L(C_z^k))$, $z\in Z_k$, $k\ge k_1(L)$.

Now, we have to prove that $E'_\psi (u_{k}^{{{\widetilde{T}}}_k^L})[v]=0$ $\forall v\in H^1_0( \Omega )$. Taking into account Lemma 3.16, we obtain

$$\begin{aligned} \begin{aligned} E'_\psi (u_{k}^{{{\widetilde{T}}}_k^L})[v]&= \int _\Omega [{\nabla }u_{k}^{{{\widetilde{T}}}_k^L}\cdot {\nabla }v-| u_{k}^{{{\widetilde{T}}}_k^L}|^{p-1} u_{k}^{{{\widetilde{T}}}_k^L}v-\psi v]\,dx\\&= \sum _{k\in Z_k} \int _{{{\widetilde{T}}}_k^L(C_z^k)}[{\nabla }u_{k}^{{{\widetilde{T}}}_k^L}\cdot {\nabla }v-| u_{k}^{{{\widetilde{T}}}_k^L}|^{p-1} u_{k}^{{{\widetilde{T}}}_k^L}v-\psi v]\,dx\\&= \sum _{k\in Z_k} \int _{\partial {{\widetilde{T}}}_k^L(C_z^k)}v\, ({\nabla }u_{k}^{{{\widetilde{T}}}_k^L}\cdot \nu _{k,z})\, d\sigma \end{aligned} \end{aligned}$$

(3.138)

where $\nu _{k,z}$ denotes the outward normal on $\partial {{\widetilde{T}}}_k^L(C_z^k)$. Thus, in order to obtain $E'_\psi (u_{k}^{{{\widetilde{T}}}_k^L})[v]=0$, we prove that

$$\begin{aligned} {\nabla }u_{k,z_1}^{{{\widetilde{T}}}_k^L}(x)={\nabla }u_{k,z_2}^{{{\widetilde{T}}}_k^L}(x)\qquad \forall x\in \partial {{\widetilde{T}}}_k^L(C_{z_1}^k)\cap \partial {{\widetilde{T}}}_k^L(C_{z_2}^k) \end{aligned}$$

(3.139)

for all $z_1,z_2\in Z_k$ such that $|z_1-z_2|=1$ (that is when ${{\widetilde{T}}}_k^L(C_{z_1}^k)$ and ${{\widetilde{T}}}_k^L(C_{z_2}^k)$ are adjacent subdomains of $\Omega$).

Notice that, since for all $k\ge k_1(L)$ and $z\in Z_k$ the function $u_{k,z}^{{{\widetilde{T}}}_k^L}$ is a solution of problem (3.115), for all vector field $\Phi \in {{\mathcal {C}}}^1_0(\overline{\Omega },{\mathbb {R}}^n)$ we obtain

$$\begin{aligned} E'_\psi ( u_{k}^{{{\widetilde{T}}}_k^L})[\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L}]= \int _\Omega [{\nabla }u_{k}^{{{\widetilde{T}}}_k^L}\cdot {\nabla }(\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L})- |u_{k}^{{{\widetilde{T}}}_k^L}|^{p-1}u_{k}^{{{\widetilde{T}}}_k^L}(\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L})-\psi (\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L})]\, dx \end{aligned}$$

$$\begin{aligned} \begin{aligned}&=\sum _{z\in Z_k}\int _{{{\widetilde{T}}}_k^L(C_z^k)} [{\nabla }u_{k}^{{{\widetilde{T}}}_k^L}\cdot {\nabla }(\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L})-|u_{k}^{{{\widetilde{T}}}_k^L}|^{p-1} u_{k}^{{{\widetilde{T}}}_k^L}(\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L})-\psi (\Phi \cdot u_{k}^{{{\widetilde{T}}}_k^L})]\,dx\\&=\sum _{z\in Z_k}\int _{\partial {{\widetilde{T}}}_k^L(C_z^k)} ({\nabla }u_{k}^{{{\widetilde{T}}}_k^L}\cdot \nu _{k,z})^2(\Phi \cdot \nu _{k,z})\,d\sigma . \end{aligned} \end{aligned}$$

(3.140)

Thus, it is easy to verify that in order to prove (3.139) it suffices to show that there exists ${\bar{k}}$ such that

$$\begin{aligned} E'_\psi (u_{k}^{{{\widetilde{T}}}_k^L})[\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L}]=0\qquad \forall \Phi \in {{\mathcal {C}}}^1_0(\overline{\Omega },{\mathbb {R}}^n),\ \forall k\ge {\bar{k}}. \end{aligned}$$

(3.141)

From Proposition 3.15 we infer that there exists ${\bar{k}}\ge k_1(L)$ such that ${{\widetilde{T}}}_k^L\in D_{L/2}$ $\forall k\ge {\bar{k}}$. Now, for all $\tau \in {\mathbb {R}}$ and $\Phi \in {{\mathcal {C}}}^1_0(\overline{\Omega },{\mathbb {R}}^n)$, let us consider the function $T_{\tau ,\Phi }:\overline{\Omega }\rightarrow \overline{\Omega }$ defined by the Cauchy problem

$$\begin{aligned} \frac{\partial T_{\tau ,\Phi }(x)}{\partial \tau }=\Phi \circ T_{\tau ,\Phi }(x),\qquad T_{0,\Phi }(x)=x\quad \forall \tau \in {\mathbb {R}},\ \forall x\in \overline{\Omega }. \end{aligned}$$

(3.142)

One can verify by standard arguments that for all $\Phi \in {{\mathcal {C}}}^1_0(\overline{\Omega },{\mathbb {R}}^n)$ there exists ${\bar{\tau }}_\Phi >0$ such that $T_{\tau ,\Phi }\circ {{\widetilde{T}}}_k^L\in D_L$ $\forall \tau \in [-{\bar{\tau }}_\Phi ,{\bar{\tau }}_\Phi ]$. It follows that

$$\begin{aligned} E_\psi \left( u_{k}^{{{\widetilde{T}}}_k^L}\right) =\sum _{z\in Z_k}E_\psi (u_{k,z}^{{{\widetilde{T}}}_k^L})\le \sum _{z\in Z_k}E_\psi \left( u_{k,z}^{T_{\tau ,\Phi }\circ {{\widetilde{T}}}_k^L}\right) = E_\psi (u_{k}^{T_{\tau ,\Phi }\circ {{\widetilde{T}}}_k^L})\qquad \forall \tau \in [-{\bar{\tau }}_\Phi ,{\bar{\tau }}_\Phi ] \nonumber \\ \end{aligned}$$

(3.143)

because of the minimality of ${{\widetilde{T}}}_k^L$. Moreover, notice that

$$\begin{aligned} \frac{d\ }{d\tau }E_\psi (u_{k}^{{{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau ,\Phi })_{|_{\tau =0}}=-E'_\psi \left( u_{k}^{{{\widetilde{T}}}_k^L}\right) [\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L}], \end{aligned}$$

(3.144)

so we have to prove that

$$\begin{aligned} \frac{d\ }{d\tau }E_\psi \left( u_{k}^{{{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau ,\Phi }\right) _{|_{\tau =0}}=0. \end{aligned}$$

(3.145)

Arguing by contradiction, assume that (3.145) does not hold. We can assume, for example, that

$$\begin{aligned} \frac{d\ }{d\tau }E_\psi (u_{k}^{{{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau ,\Phi })_{|_{\tau =0}}<0 \end{aligned}$$

(3.146)

(otherwise we replace $\Phi$ by $-\Phi$). Therefore, there exists a sequence of positive numbers $(\tau _i)_i$ such that $\lim _{i\rightarrow \infty }\tau _i=0$ and

$$\begin{aligned} E_\psi (u_{k}^{{{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau _i,\Phi })<E_\psi ({\tilde{u}}_{k}^{{{\widetilde{T}}}_k^L})\qquad \forall i\in {\mathbb {N}}. \end{aligned}$$

(3.147)

We say that, as a consequence of (3.146), for i large enough we have

$$\begin{aligned} \max \left\{ \sum _{z\in Z_k}E_\psi \big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi }+t_z(u_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi }-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi })\big ) \,\ t_z\ge 0\ \forall z\in Z_k\right\} <E_\psi (u_{k}^{{{\widetilde{T}}}_k^L}).\nonumber \\ \end{aligned}$$

(3.148)

In fact, arguing by contradiction, assume that (up to a subsequence still denoted by $(\tau _i)_i$) the inequality (3.148) does not hold.

Then, for all $i\in {\mathbb {N}}$ and $z\in Z_k$, there exists $t_{z,i}\ge 0$ such that

$$\begin{aligned} \sum _{z\in Z_k}E_\psi \big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi }+t_{z,i} (u_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi }-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi })\big )\ge E_\psi (u_{k}^{{{\widetilde{T}}}_k^L})\qquad \forall i\in {\mathbb {N}}.\nonumber \\ \end{aligned}$$

(3.149)

Since $p>1$, the sequence $(t_{z,i})_i$ is bounded $\forall z\in Z_k$. Moreover, taking into account that

$$\begin{aligned} E_\psi ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} +t (u_{k,z}^{{{\widetilde{T}}}_k^L} -{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L})\big )<E_\psi (u_{k,z}^{{{\widetilde{T}}}_k^L})\qquad \forall t\ne 1,\ \forall z\in Z_k, \end{aligned}$$

(3.150)

we infer that $\lim _{i\rightarrow \infty }t_{z,i}=1$ $\forall z\in Z_k$ and

$$\begin{aligned} \begin{aligned} \sum _{z\in Z_k}E_\psi \big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi }+t_{z,i} (u_{k,z}^{{{\widetilde{T}}}_k^L}&\circ T^{-1}_{\tau _i,\Phi }-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi })\big ) \ge E_\psi (u_{k}^{{{\widetilde{T}}}_k^L})\\&\ge \sum _{z\in Z_k}E_\psi \big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} +t_{z,i} (u_{k,z}^{{{\widetilde{T}}}_k^L} -{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} )\big )\qquad \forall i\in {\mathbb {N}}. \end{aligned} \end{aligned}$$

(3.151)

As a consequence, for all $i\in {\mathbb {N}}$ there exists $\tau '_i\in ]0,\tau _i[$ such that

$$\begin{aligned} \frac{d\ }{d\tau } \sum _{z\in Z_k}E_\psi \big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau ,\Phi }+t_{z,i} (u_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau ,\Phi }-{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau ,\Phi })\big )_{|_{\tau =\tau _i'}}\ge 0\qquad \forall i\in {\mathbb {N}}\end{aligned}$$

(3.152)

which, as $i\rightarrow \infty$, implies

$$\begin{aligned} \frac{d\ }{d\tau } E_\psi (u_{k}^{{{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau ,\Phi })_{|_{\tau = 0}}\ge 0, \end{aligned}$$

(3.153)

in contradiction with (3.146). Thus, (3.148) holds. From Lemma 2.4 we infer that, if we choose ${\bar{k}}$ large enough, for all $k\ge {\bar{k}}$, $z\in Z_k$ and $i\in {\mathbb {N}}$ there exists a unique minimizing function ${\tilde{u}}_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}$. Moreover, ${\tilde{u}}_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}\rightarrow {\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L}$ in $H^1_0(\Omega )$, as $i\rightarrow \infty$, $\forall k\ge {\bar{k}}$, $\forall z\in Z_k$.

Then, using the functions ${\tilde{u}}_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}$ and arguing as in the proof of Lemma 3.16, for i large enough we obtain the functions $u_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}$.

The construction of the functions ${\tilde{u}}_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}$ and $u_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}$ shows also that

$$\begin{aligned} E_\psi ( {\tilde{u}}_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L})\le E_\psi ( {\tilde{u}}_{k,z}^{ {{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau _i,\Phi }) \end{aligned}$$

(3.154)

and

$$\begin{aligned} E_\psi ( u_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L})\le \max \left\{ E_\psi ({\tilde{u}}_{k,z}^{ {{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau _i,\Phi }+t\big ( u_{k,z}^{ {{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau _i,\Phi }-{\tilde{u}}_{k,z}^{ {{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau _i,\Phi } )\big )\,\ t\ge 0\right\} \qquad \forall z\in Z_k.\nonumber \\ \end{aligned}$$

(3.155)

Therefore, from (3.148) and (3.155) we obtain

$$\begin{aligned} E_\psi (u_{k}^{{{\widetilde{T}}}_k^L})> & {} \max \left\{ \sum _{z\in Z_k}E_\psi \big ({\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi }+t_{z} (u_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi } -{\tilde{u}}_{k,z}^{{{\widetilde{T}}}_k^L} \circ T^{-1}_{\tau _i,\Phi })\big )\ t_z\ge 0\ \forall z\in Z_k \right\} \nonumber \\\ge & {} \sum _{z\in Z_k}E_\psi \big (u_{k,z}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}\big ) =E_\psi \big (u_{k}^{T_{\tau _i,\Phi }\circ {{\widetilde{T}}}_k^L}\big ) \end{aligned}$$

(3.156)

for i large enough, in contradiction with (3.143).

So we can conclude that $\frac{d\ }{d\tau }E_\psi \left( u_{k}^{{{\widetilde{T}}}_k^L}\circ T^{-1}_{\tau _i,\Phi }\right) =0$, that is $E'_\psi (u_{k}^{{{\widetilde{T}}}_k^L})[\Phi \cdot {\nabla }u_{k}^{{{\widetilde{T}}}_k^L}]=0$ for all vector field $\Phi \in {{\mathcal {C}}}^1_0(\overline{\Omega },{\mathbb {R}}^n)$.

Thus, $u_{k}^{{{\widetilde{T}}}_k^L}$ is a solution of problem (1.1) for all $k\ge {\bar{k}}$. $\square$

Proof of Proposition 2.2 (conclusion)

If $\Omega$ is the cube (2.1), all the assertions of Proposition 2.2 hold for k large enough if we set $u_k=u_{k}^{{{\widetilde{T}}}_k^L}$ and $T_{k,u}=T^L_k$ (or $T_{k,u}={{\widetilde{T}}}_k^L$) where the function $u_k^{T_k^L}$ and the admissible deformation $T^L_k$ are obtained by the minimizing method described in Sect. 2 (the functions $v_k=v_k^{T_{k,v}}$ are obtained in a similar way: it suffices to replace $\sigma (z)$ by $\sigma (z)+1$). Notice that we have $u_k^{T^L_k}=u_k^{{{\widetilde{T}}}_k^L}$ (where ${{\widetilde{T}}}_k^L:\overline{\Omega }\rightarrow \overline{\Omega }$ is the function introduced in Definition 3.14) because $T^L_k(C^k_z)={{\widetilde{T}}}^L_k(C^k_z)$ $\forall z\in Z_k$.

In fact, Proposition 3.17 guarantees that there exists ${\bar{k}}\in {\mathbb {N}}$ such that $u_k^{{{\widetilde{T}}}_k^L}$ is a solution of problem (1.1) for all $k\ge {\bar{k}}$.

The asymptotic behaviour of $u_k$ as $k\rightarrow \infty$ is described by Proposition 3.5, Lemma 3.7 and Proposition 3.15. In fact, Proposition 3.5 shows that for every choice of $z^k$ in $Z_k$, up to a subsequence, the function $U_{z^k}$, as $k\rightarrow \infty$ converges in $H^1(\chi )$ to a positive solution $U_\chi$ of the Dirichlet problem $-\Delta U=|U|^{p-1}U$ in $\chi$, $U=0$ on $\partial \chi$, satisfying

$$\begin{aligned} \left( \int _\chi |U_\chi |^{p+1}dx\right) ^{-\frac{2}{p+1}}\int _\chi |{\nabla }U_\chi |^2dx=\min \left\{ \int _\chi |{\nabla }U|^2dx\,\ U\in H^1_0(\chi ),\ \int _\chi |U|^{p+1}dx=1\right\} \nonumber \\ \end{aligned}$$

(3.157)

where $\chi$ is a bounded domain of ${\mathbb {R}}^n$. Lemma 3.7 says that the minimality of the admissible deformation $T^L_k$ implies that $\chi$ must be a cube of ${\mathbb {R}}^n$ having a vertex in the origin and the sides of length 1 and finally Proposition 3.15 guarantees that ${{\widetilde{T}}}_k^L\in D_{L_k}$ for a suitable sequence $(L_k)_k$ in ]0, 1[ such that $\lim _{k\rightarrow \infty }L_k=0$ so, as a consequence, $\chi =C_0^1=\Omega$ and ${{\widetilde{T}}}_k^L$ converges as $k\rightarrow \infty$ to the identity function uniformly in $\Omega$ (as pointed out in Remark 3.4).

Notice that $T^L_k\in D_L$ $\forall k\in {\mathbb {N}}$ but, unlike ${{\widetilde{T}}}_k^L$, we cannot say that $T^L_k\in D_{L_k}$ $\forall k\in {\mathbb {N}}$. However, we can say that also $T^L_k$ converges to the identity function uniformly in $\Omega$ because, taking into account the definition of ${{\widetilde{T}}}_k^L$, we have

$$\begin{aligned} T^L_k(x)\in {{\widetilde{T}}}_k^L(C_z^k)\qquad \forall x\in C^k_z,\ \forall z\in Z_k \end{aligned}$$

(3.158)

so, as a consequence,

$$\begin{aligned} \sup \{|T_k^L(x)-{{\widetilde{T}}}_k^L(x)|\,\ x\in \Omega \}\le (1+L)\frac{\sqrt{n}}{k}. \end{aligned}$$

(3.159)

Therefore, all the assertions in Proposition 2.2 hold for $T_{k,u}={{\widetilde{T}}}_k^L$ and also for $T_{k,u}=T^L_k$, so the proof is complete. $\square$

Theorem 2.1 is a direct consequence of Proposition 2.2.

Notice that our method to construct solutions having this checked nodal structure does not require any technique of deformation from the symmetry and it works in case of more general nonlinearities, even when they are not perturbations of symmetric nonlinearities by lower order terms.

For example, it works when in problem (1.1) the term $|u|^{p-1}u+\psi$ is replaced by $c_+(u^+)^p-c_-(u^-)^p+\psi$ with $c_+>0$, $c_->0$ and $c_+\ne c_-$.

Moreover, notice that our method works also when the nonlinear term has critical growth. For example, for $n>2$ and $\lambda \in {\mathbb {R}}$ let us consider the Dirichlet problem

$$\begin{aligned} -\Delta u=|u|^{\frac{4}{n-2}}u+\lambda u+\psi \quad \text{ in } \Omega ,\qquad u=0\quad \text{ on } \partial \Omega \end{aligned}$$

(3.160)

whose solutions are critical points of the energy functional ${{\mathcal {F}}}:H^1_0(\Omega )\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} {{\mathcal {F}}}(u)=\frac{1}{2}\int _\Omega |{\nabla }u|^2dx-\frac{n-2}{2n}\int _\Omega |u|^{\frac{2n}{n-2}}dx-\frac{\lambda }{2}\int _\Omega u^2 dx-\int _\Omega \psi u\, dx. \end{aligned}$$

(3.161)

It is well known that, if $\lambda =0$ and $\psi \equiv 0$ in $\Omega$, problem (3.160) has only the trivial solution $u\equiv 0$ for every bounded starshaped domain $\Omega$, as a consequence of the Pohozaev identity (see [36]).

When $n\ge 4$, $\psi \equiv 0$ in $\Omega$ and $\lambda$ is positive and strictly less than the first eigenvalue of the Laplace operator $-\Delta$ in $H^1_0(\Omega )$, there exists a positive solution that concentrates as a Dirach mass as $\lambda \rightarrow 0$ (see [12, 13] etc.); the existence of nodal solutions is studied for example in [16] etc..

Notice that also when $\psi \equiv 0$ in $\Omega$, so that the functional ${{\mathcal {F}}}$ is even, the problem of finding infinitely many solutions is difficult because the well known Palais-Smale compactness condition is not satisfied, as a consequence of the presence of the critical Sobolev exponent (see [12, 13, 44] etc.).

When $\Omega$ is a cube of ${\mathbb {R}}^n$, our method, combined with some estimates as in [13], allows us to construct infinitely many solutions with many nodal regions and arbitrarily large energy level for all $\lambda >0$ and $\psi \in L^2(\Omega )$.

In fact, as we prove in a paper in preparation, the following theorem holds (see also [27] for the particular case where $\Omega$ is a cube and $\psi \equiv 0$ in $\Omega$).

Theorem 3.18

Assume that $\Omega$ is a cube of ${\mathbb {R}}^n$ with $n\ge 4$ and $\lambda >0$. Then, for every $\psi \in L^2(\Omega )$, problem (3.160) admits infinitely many solutions.

More precisely, if $\Omega$ is for example the cube (2.1), for all $\psi \in L^2(\Omega )$ there exists ${\bar{k}}\in {\mathbb {N}}$ such that, for every $k\ge {\bar{k}}$, problem (3.160) admits a solution $u_k$ having the following properties.

For all $k\ge {\bar{k}}$ there exists $T_k\in D_L$ such that, for every choice of $z^k$ in $Z_k$, the function $u_{k,z^k}:={u_k}_{|_{T_k(C^k_{z^k})}}$ belongs to $H^1_0\big (T_k(C^k_{z^k})\big )$ (here we consider $u_{k,z^k}$ extended by the value zero in ${\mathbb {R}}^n\setminus \Omega$).

Moreover, there exist ${\varepsilon }_k>0$ and $m_k\in C^1_0=\Omega$ such that ${\varepsilon }_k\rightarrow 0$ as $k\rightarrow \infty$ and the function $U_k$ defined by

$$\begin{aligned} U_k(x)=\sigma (z^k)\left( \frac{{\varepsilon }_k}{k}\right) ^{\frac{n-2}{2}}u_{k,z^k}\left[ \frac{{\varepsilon }_k}{k}\,x +T_k\left( \frac{z^k+m_k}{k}\right) \right] \qquad \forall x\in {\mathbb {R}}^n,\ \forall k\in {\mathbb {N}}\end{aligned}$$

(3.162)

converges as $k\rightarrow \infty$ to a function ${\overline{U}}\in {{\mathcal {D}}}({\mathbb {R}}^n)$ such that

$$\begin{aligned} -\Delta {\overline{U}}={\overline{U}}^{\frac{n+2}{n-2}},\qquad {\overline{U}}>0\ \text{ in } {\mathbb {R}}^n,\qquad {\overline{U}}(0)=\max _{{\mathbb {R}}^n}{\overline{U}}. \end{aligned}$$

(3.163)

The sequence $(T_k)_k$ converges to the identity map uniformly in $\Omega$ while the domains $k\Big [T_k(C_{z^k}^k)$ $-T_k\left( \frac{z^k}{k}\right) \Big ]$ tend to the cube $\Omega$ as $k\rightarrow \infty$ for every choice of $z^k$ in $Z_k$.

Furthermore, for all $k\ge {\bar{k}}$ there exists also another solution $v_k$ of problem (3.160) such that the function $-v_k$ presents an asymptotic behaviour as $u_k$ when $k\rightarrow \infty$.