Infinitely many solutions for elliptic equations with non-symmetric nonlinearities

Abstract

We deal with the existence of infinitely many solutions for a class of elliptic problems with non-symmetric nonlinearities. Our result, which is motivated by a well known conjecture formulated by A. Bahri and P.L. Lions, suggests a new approach to tackle these problems. The proof is based on a method which does not require to use techniques of deformation from the symmetry and may be applied to more general non-symmetric problems.

1 Introduction

Let us consider the problem

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

(1.1)

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

If $w\not \equiv 0$ in $\Omega$, the corresponding energy functional $E:H^1_0(\Omega )\rightarrow \mathbb {R}$, defined by

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

(1.2)

is not even, so the equivariant Lusternik-Schnirelmann theory for $\mathbb {Z}_2$-symmetric sets cannot be applied to find infinitely many solutions as in the case $w\equiv 0$ (see for instance [1, 3, 18, 19, 27, 28, 32, 34] and also [9, 17] for a more general framework).

In the case $w\not \equiv 0$ in $\Omega$, a natural question (which goes back to the beginning of the eighties) is wether the infinite number of solutions still persists under perturbation.

A detailed analysis was originally carried on in [2, 3, 5,6,7,8, 25, 29, 30, 33, 35, 39] by Ambrosetti, Bahri, Berestycki, Ekeland, Ghoussoub, Krasnoselskii, Lions, Marino, Prodi, Rabinowitz, Struwe and Tanaka by introducing new perturbation methods. In particular, this question was raised to the attention by Rabinowitz also in his monograph on minimax methods (see [34, 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 w 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 $w\in L^2(\Omega )$.

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 $1<p<\frac{n+2}{n-2}$.

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, 15], which include also applications to more general nonlinear problems). However that approach did not allow to solve the Bahri-Lions conjecture.

In the present paper we describe a new possible method to approach this problem. The idea is to trying to piece together solutions of Dirichlet problems in subdomains of $\Omega$ chosen in a suitable way. This idea has been first used by Struwe in earlier papers (see [35,36,37] and references therein). In the present paper we consider as nodal regions subdomains of $\Omega$ that are deformations of cubes by suitable Lipschitz maps (we say that nodal solutions of this type have a check structure).

It is interesting to observe that such a class of Lipshitz maps also appeared in some recent works by Rabinowitz and Byeon (see [13, 14] and the references therein) concerning a rather different problem: construct solutions having certain prescribed patterns for an Allen-Cahn model equation. Also in that papers, as in the present one, Lipschitz condition is combined with the structure of $\mathbb {Z}^n$ and the covering of $\mathbb {R}^n$ by cubes with vertices in $\mathbb {Z}^n$.

The main multiplicity result of this paper is stated in Theorem 2.7 and says that if $\Omega$ satisfies a suitable geometric condition (condition (2.53)) then problem (1.1) admits infinitely many nodal solutions having as nodal structure suitable partitions of $\Omega$ in subdomains that are Lipschitz deformations of arbitrarily small cubes.

In order to explain the meaning of the geometrical condition (2.53), which plays a crucial role in this paper, we need some more detail on the method we use in the proof. Let $P_k$ denotes 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 $L>1$, consider the class $\mathcal{C}_L(P_k,{\overline{\Omega }})$ consisting of the bilipschitz maps between $P_k$ and ${\overline{\Omega }}$ with Lipshitz constants in $\left[ \frac{1}{L}, L\right]$. For all T in $\mathcal{C}_L(P_k,{\overline{\Omega }})$ and k in $\mathbb {N}$, we construct by a mini-max argument a nodal function $u_k^T$ in $H^1_0(\Omega )$, whose nodal regions are the deformations of these cubes by the map T. For k large enough, $u^T_k$ satisfies equation (1.1) in each nodal region (by Proposition 2.4) and it is solution of the Dirichlet problem (1.1) when, in addition, it satisfies the assumptions of Proposition 2.5 (a kind of stationarity property). Then, for all k, we minimize the energy functional E in the set $\{u^T_k\ :\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\}$ and show that, if the minimum is achieved by a bilipschitz map $T_k^L$ with Lipschitz constants in the interior of $\left[ \frac{1}{L}, L\right]$, then the corresponding function $u_k^{T^L_k}$ satisfies the stationarity condition of Proposition 2.5, so it is a solution of Problem (1.1) for k large enough.

Condition (2.53) requires just that, for a suitable choice of $L>1$, there exists ${\bar{k}}$ in $\mathbb {N}$ and ${\bar{L}}$ in ]1, L[ such that the minimum is achieved by a map $T^L_k$ in $\mathcal{C}_{\bar{L}}(T_k,{\overline{\Omega }})$ for all positive integer $k\ge {\bar{k}}$.

Thus, if $\Omega$ satisfies condition (2.53), we obtain infinitely many nodal solutions $u^{T^L_k}_k$ having check structure. Moreover, the number of nodal regions of $u_k^{T^L_k}$ and its energy $E(u_k^{T^L_k})$ tend to infinity as $k\rightarrow \infty$, while the size of the nodal regions tends to zero.

Lemma 2.9 and Corollary 2.10 show that condition (2.53) holds true, for example, when $n=1$ (the proof may be also adapted to deal with radial solutions in domains $\Omega$ having radial symmetry).

Indeed, in dimension $n=1$, a more general result was obtained by Ehrmann in [23] (see also [24, 26] for related results). Here it is proved that the ordinary differential equation

$$\begin{aligned} -u''(x)=f(u(x))+w(x)\qquad \text { for }x\in (0,1),\ u(0)=u(1)=0 \end{aligned}$$

(1.3)

has infinitely many distinct solutions when f is a function with superlinear growth satisfying quite general assumptions. However, the method here used relies on a shooting argument, typical of ordinary differential equations, combined with counting the oscillations of the solutions in the interval (0, 1). Therefore, this method, which gives the existence of solutions having a sufficiently large number of zeroes in dimension $n=1$, cannot be extended to higher dimensions.

On the contrary, in the present paper we use a method which is more similar to the one introduced by Nehari in [31], that can be in a natural way extended to the case $n>1$. In fact, for example, Nehari’s work was followed up by Coffman who studied an analogous problem for partial differential equations (see [18, 19]). Independently, this problem was also studied by Hempel (see [27, 28]).

More recently, the method introduced by Nehari has been also used by Conti, Terracini and Verzini to study optimal partition problems in n-dimensional domains and related problems: in particular, existence of minimal partitions and extremality conditions, behaviour of competing species systems with large interactions, existence of changing sign solutions for superlinear elliptic equations, etc. (see [20,21,22, 40]).

Notice that Nehari’s work deals with an odd differential operator, so the corresponding energy functional is even. Moreover, Nehari proves that for every positive integer k there exists a solution having exactly k zeroes. On the contrary, in the present paper (as Ehrmann in [23]) we find only solutions with a large number of zeroes; moreover, we prove that, for all w in $L^2(\Omega )$, the zeroes tend to be uniformly distributed in all of the domain $\Omega$ as their number tends to infinity (see Lemmas 2.9 and 3.2). The reason is that, when $w\not \equiv 0$, the Nehari type argument we use in the proof works only when the sizes of all the nodal regions are small enough, so their number is sufficiently large.

In order to show that our existence result is sharp, we prove also that the term w in problem (1.1) can be chosen in such a way that the problem does not have solutions with a small number of nodal regions. More precisely, in the case $n=1$ we show that for all positive integer h there exists $w_h$ in $L^2(\Omega )$ such that every solution of problem (1.1) with $w=w_h$ has at least h zeroes (see Corollary 3.6). Indeed, we show that for all $n\ge 1$ and for every eigenfunction $e_k$ of the Laplace operator $-\Delta$ in $H^1_0(\Omega )$ there exists ${\bar{w}}_k$ in $L^2(\Omega )$ such that every solution u of problem (1.1) with $w={\bar{w}}_k$ must have the sign related to the sign of $e_k$ in the sense that every nodal region of $e_k$ has a subset where u and $e_k$ have the same sign (see Proposition 3.5).

In the case $n>1$, condition (2.53) seems to be more difficult to be verified because the class $\mathcal{C}_L(P_k,{\overline{\Omega }})$ of the admissible deformations of the nodal structure might result too large. In fact, as we point out in Remark 3.1, the minimality of the map $T^L_k$ implies that as $k\rightarrow \infty$ the nodal regions of $u_k^{T^L_k}$ tend to have all the same size. In the case $n=1$, this property is sufficient to control the Lipschitz constant of $T^L_k$, for k large enough, in order to prove condition (2.53). On the contrary, if $n>1$ the Lipschitz constant of $T^L_k$ might be very large, even if the nodal regions of $u_k^{T^L_k}$ tend as $k\rightarrow \infty$ to have all the same size, because their shape might be very different from the cubes of $\mathbb {R}^n$. Therefore, a natural idea is to restrict the class of admissible deformations $\mathcal{C}_L(P_k,{\overline{\Omega }})$, taking also into account that our method to construct nodal solutions with many small nodal regions having check structure can be easily adapted to deal with other classes of bilipschitz maps, different from $\mathcal{C}_L(P_k,\overline{\Omega })$.

For example, we can fix a bilipschitz map $T_0:\overline{\Omega }\rightarrow {\overline{\Omega }}$ and consider as a class of admissible deformations a suitable neighbourhood of $T_0$ in $\mathcal{C}_L(P_k,{\overline{\Omega }})$, that is the class of all the maps in $\mathcal{C}_L(P_k,{\overline{\Omega }})$ that are close to $T_0$ in a suitable sense (see Remark 3.1). Then, the geometrical condition (2.53) has to be replaced by a similar condition that holds or fails depending on the choice of $T_0$ and of its neighborhood (condition (3.7)) whose meaning is again that, for k large enough, the minimization problem settled in this new class of admissible deformations is achieved by a map which is, in some sense, in the interior of this new class. Thus, if $n>1$, the problem is to choose carefully a suitable class of admissible deformations.

In a similar way, for example, we can prove that if $\Omega$ is a cube of $\mathbb {R}^n$ with $n>1$, $p>1$, $p<\frac{n+2}{n-2}$ if $n>2$, for all w in $L^2(\Omega )$ there exist infinitely many solutions $u_k(x)$ of problem (1.1) such that the nodal regions of the function $u_k\left( \frac{x}{k}\right)$, after translations, tend to the cube as $k\rightarrow \infty$ (the proof will be reported in a paper in preparation).

Notice that, in particular, this result shows that the Bahri-Lions conjecture is true when $\Omega$ is a cube of $\mathbb {R}^n$ with $n\ge 3$.

Let us point out that our method does not require techniques of deformation from the symmetry and may be applied to more general problems: for example, when the nonlinear term $|u|^{p-1}u$ is replaced by $c_+(u^+)^p-c_-(u^-)^p$ with $c_+$ and $c_-$ two positive constants (see Lemma 3.2), in case of different, nonhomogeneous boundary conditions and even in case of nonlinear elliptic equations involving critical Sobolev exponents.

2 Existence of infinitely many nodal solutions

In order to find infinitely many solutions with an arbitrarily large number of nodal regions, we proceed as follows.

Let us set

$$\begin{aligned}&\begin{array}{c} C_0=\{x\in \mathbb {R}^n\ :\ 0<x_i<1\ \text{ for } i=1,\ldots ,n\},\\ C_z=z+C_0,\quad \sigma (z)=(-1)^{\sum _{i=1}^n z_i}\qquad \forall z\in \mathbb {Z}^n, \end{array} \end{aligned}$$

(2.1)

$$\begin{aligned}&Z_k=\left\{ z\in \mathbb {Z}^n\ :\ {1\over k}\, C_z\subset \Omega \right\} ,\qquad P_k=\bigcup _{z\in Z_k}{1\over k}\, {\overline{C}}_z,\qquad \forall k\in \mathbb {N}. \end{aligned}$$

(2.2)

Notice that there exists $k_\Omega$ in $\mathbb {N}$ such that $Z_k\ne \emptyset$ $\forall k\ge k_\Omega$.

For all subsets P, Q of $\mathbb {R}^n$ and for all $L\ge 1$, let us denote by $\mathcal{C}_L(P,Q)$ the set of all the functions $T:P\rightarrow Q$ such that

$$\begin{aligned} {1\over L}\,|x-y|\le |T(x)-T(y)|\le L\, |x-y|\qquad \forall x,y\in P. \end{aligned}$$

(2.3)

For all $k\ge k_\Omega$, $z\in Z_k$, $L\ge 1$, $T\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ let us set

$$\begin{aligned} E^T_{k,z}=\inf \left\{ E(u)\ :\ u\in H^1_0\left( T\left( {1\over k}\, C_z\right) \right) ,\ \int _{T\left( {1\over k}\, C_z\right) }|u|^{p+1}dx=1\right\} . \end{aligned}$$

(2.4)

Since $p<{n+2\over n-2}$ when $n\ge 3$, one can easily verify that the infimum in (2.4) is achieved. Moreover, for all $L\ge 1$ and $k\ge k_\Omega$, also the infimum

$$\begin{aligned} \inf \{E^T_{k,z}\ :\ z\in Z_k,\ T\in C_L(P_k,{\overline{\Omega }})\} \end{aligned}$$

(2.5)

is achieved (as one can prove by standard arguments using Ascoli-Arzelà Theorem) and the following lemma holds.

Lemma 2.1

For all $L\ge 1$, we have

$$\begin{aligned} \lim _{k\rightarrow \infty }\min \{E ^T_{k,z}\ :\ z\in Z_k,\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\}=\infty . \end{aligned}$$

(2.6)

Proof

For all $L\ge 1$ and $k\ge k_\Omega$, let us choose $z_k\in Z_k$, $T_k\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ and ${\bar{u}}_k\in H^1_0\left( T_k\left( {1\over k}C_{z_k}\right) \right)$ such that

$$\begin{aligned} \int _{T_k\left( {1\over k}C_{z_k}\right) }|\bar{u}_k|^{p+1}dx=1 \ \text{ and } \ E(\bar{u}_k)=E^{T_k}_{k,z_k}=\min \{E^T_{k,z}\ :\ z\in Z_k,\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\}.\nonumber \\ \end{aligned}$$

(2.7)

We say that

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{T_k({1\over k}\,C_{z_k})}|{\nabla }{\bar{u}}_k|^{2}dx=\infty . \end{aligned}$$

(2.8)

In fact, arguing by contradiction, assume that

$$\begin{aligned} \liminf _{k\rightarrow \infty }\int _{T_k({1\over k}\,C_{z_k})}|{\nabla }\bar{u}_k|^{2}dx<\infty . \end{aligned}$$

(2.9)

It follows that (up to a subsequence) $({\bar{u}}_k)_k$ is bounded in $H^1_0(\Omega )$ and there exists a function ${\bar{u}}\in H^1_0(\Omega )$ such that ${\bar{u}}_k\rightarrow {\bar{u}}$, as $k\rightarrow \infty$, weakly in $H^1_0(\Omega )$, in $L^{p+1}(\Omega )$, and almost everywhere in $\Omega$ (here ${\bar{u}}_k$ is extended by the value 0 in $\Omega \setminus T_k({1\over k}\,C_{z_k})$). Since $\mathop {\mathrm{meas}}\nolimits \left( T_k({1\over k}\,C_{z_k})\right) \rightarrow 0$ as $k\rightarrow \infty$, from the almost everywhere convergence we obtain ${\bar{u}}\equiv 0$ in $\Omega$, which is a contradiction because ${\bar{u}}_k\rightarrow {\bar{u}}$ in $L^{p+1}(\Omega )$ and (2.7) holds for all $k\ge k_\Omega$. Thus (2.8) is proved.

Notice that

$$\begin{aligned} E^{T_k}_{k,z_k}={1\over 2}\int _{T_k\left( {1\over k}\, C_{z_k}\right) }|{\nabla }{\bar{u}}_k|^2dx-{1\over p+1}- \int _{T_k\left( {1\over k}\, C_{z_k}\right) } {\bar{u}}_kw\, dx\qquad \forall k\ge k_\Omega \end{aligned}$$

(2.10)

where

$$\begin{aligned} \begin{array}{rcl} \left| \int _{T_k({1\over k}\,C_{z_k})} {\bar{u}}_k\, w\, dx \right| &{} \le &{} \left( \int _{T_k({1\over k}\,C_{z_k})}{\bar{u}}_k^2 dx\right) ^{1\over 2} \left( \int _{T_k({1\over k}\,C_{z_k})} w^2 dx\right) ^{1\over 2} \\[2ex] &{} \le &{} \left[ \mathop {\mathrm{meas}}\nolimits \left( T_k({1\over k}\,C_{z_k})\right) \right] ^{{1\over 2}-{1\over p+1}}\left( \int _{\Omega } w^2 dx\right) ^{1\over 2}. \end{array} \end{aligned}$$

(2.11)

As a consequence, for all $k\ge k_\Omega$ we obtain

$$\begin{aligned} E^{T_k}_{k,z_k}\ge {1\over 2} \int _{T_k({1\over k}\,C_{z_k})}|{\nabla }{\bar{u}}_k|^2 dx-{1\over p+1}- \left( \int _{\Omega } w^2 dx\right) ^{1\over 2}\cdot \left[ \mathop {\mathrm{meas}}\nolimits \left( T_k\left( {1\over k}\,C_{z_k}\right) \right) \right] ^{{1\over 2}-{1\over p+1}},\nonumber \\ \end{aligned}$$

(2.12)

and, as $k\rightarrow \infty$,

$$\begin{aligned} \lim _{k\rightarrow \infty }E^{T_k}_{k,z_k}=\infty \end{aligned}$$

(2.13)

which completes the proof. $\square$

Corollary 2.2

For all $L\ge 1$ there exists $k(L)\ge k_\Omega$ such that for all $k\ge k(L)$, $z\in Z_k$ and $T\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ the minimum

$$\begin{aligned} \min \left\{ E(u)\ :\ u\in H^1_0\left( T\left( {1\over k}\, C_z\right) \right) ,\ \int _{T\left( {1\over k}\, C_z\right) }|u|^{p+1}dx<1\right\} \end{aligned}$$

(2.14)

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

$$\begin{aligned} \lim _{k\rightarrow \infty }\sup \left\{ \int _{T\left( {1\over k}\, C_z\right) }|{\nabla }{\tilde{u}}^T_{k,z}|^2dx\ :\ z\in Z_k,\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\right\} =0. \end{aligned}$$

(2.15)

Proof

As a consequence of Lemma 2.1, for all $L\ge 1$ there exists $k(L)\ge k_\Omega$ such that

$$\begin{aligned} 0<\min \left\{ E(u)\ :\ u\in H^1_0\left( T\left( {1\over k}\, C_z\right) \right) ,\ \int _{T\left( {1\over k}\, C_z\right) }|u|^{p+1}dx=1\right\} \end{aligned}$$

$$\begin{aligned} \forall k\ge k(L),\ \forall z\in Z_k,\ \forall T\in \mathcal{C}_L(P_k,{\overline{\Omega }}). \end{aligned}$$

(2.16)

On the other hand,

$$\begin{aligned} \inf \left\{ E(u)\ :\ u\in H^1_0\left( T\left( {1\over k}\, C_z\right) \right) ,\ \int _{T\left( {1\over k}\, C_z\right) }|u|^{p+1}dx<1\right\} \le 0 \end{aligned}$$

$$\begin{aligned} \forall k\ge k_\Omega ,\ \forall z\in Z_k,\ \forall T\in \mathcal{C}_L(P_k,{\overline{\Omega }}) \end{aligned}$$

(2.17)

because $E(u)=0$ for $u\equiv 0$ in $T\left( {1\over k}\, C_z\right)$.

Now, let us consider a minimizing sequence for the infimum in (2.17). Since it is bounded in $L^{p+1}\left( {1\over k}\, C_z\right)$, we infer from (2.17) that it is bounded also in $H^1_0\left( {1\over k}\, C_z\right)$. Therefore, since $p<{n+2\over n-2}$ when $n\ge 3$, one can prove by standard arguments that (up to a subsequence) it converges to a function ${\tilde{u}}^T_{k,z}\in H^1_0\left( T\left( {1\over k}\, C_z\right) \right)$ such that $\int _{T\left( {1\over k}\, C_z\right) }|{\tilde{u}}^T_{k,z}|^{p+1}dx$ $<1$ and

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

(2.18)

In order to prove (2.15) we argue by contradiction and assume that

$$\begin{aligned} \limsup _{k\rightarrow \infty }\sup \left\{ \int _{T\left( {1\over k}\, C_z\right) } |{\nabla }{\tilde{u}}^T_{k,z}|^2dx\ :\ z\in Z_k,\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\right\} >0. \end{aligned}$$

(2.19)

Then, for all $k\ge k(L)$ there exist $z_k\in Z_k$ and $T_k\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ such that (up to a subsequence)

$$\begin{aligned} \lim _{k\rightarrow \infty } \int _{T_k\left( {1\over k}\, C_{z_k}\right) } |{\nabla }{\tilde{u}}^{T_k}_{k,z_k}|^2dx>0. \end{aligned}$$

(2.20)

Since $E({\tilde{u}}^{T_k}_{k,z_k})\le 0$ and the sequence $\tilde{u}^{T_k}_{k,z_k}$ (extended by the value zero outside ${1\over k}\, C_{z_k}$) is bounded in $L^{p+1}(\Omega )$, we infer that it is bounded also in $H^1_0(\Omega )$. We say that, as a consequence, ${\tilde{u}}^{T_k}_{k,z_k}\rightarrow 0$ as $k\rightarrow \infty$ in $L^{p+1}(\Omega )$. In fact, since the sequence $\left( \tilde{u}^{T_k}_{k,z_k}\right) _{k\in \mathbb {N}}$ is bounded in $H^1_0(\Omega )$, it converges weakly in $H^1_0(\Omega )$, in $L^{p+1}(\Omega )$ and a.e. in $\Omega$ to a function ${\tilde{u}}\in H^1_0(\Omega )$. Since $\lim \limits _{k\rightarrow \infty }\mathop {\mathrm{meas}}\nolimits \left( {1\over k}\, C_{z_k}\right) =0$, we can say that ${\tilde{u}}\equiv 0$ in $\Omega$. Thus, $\tilde{u}^{T_k}_{k,z_k}\rightarrow 0$ as $k\rightarrow \infty$ in $L^{p+1}(\Omega )$. Therefore, taking into account that

$$\begin{aligned} E\left( {\tilde{u}}^{T_k}_{k,z_k}\right) ={1\over 2}\int _\Omega |{\nabla }{\tilde{u}}^{T_k}_{k,z_k}|^2dx-{1\over p+1} \int _\Omega |\tilde{u}^{T_k}_{k,z_k}|^{p+1} dx - \int _\Omega w \, \tilde{u}^{T_k}_{k,z_k}\, dx\le 0\nonumber \\ \end{aligned}$$

(2.21)

it follows that ${\tilde{u}}^{T_k}_{k,z_k}\rightarrow 0$ also in $H^1_0(\Omega )$ in contradiction with (2.20).

Thus, we can conclude that (2.15) holds. Finally, notice that ${\tilde{u}}^{T}_{k,z}$ is the unique minimizing function for (2.14) because the functional E is strictly convex in a suitable neighborhood of zero. So the proof is complete. $\square$

Taking into account Corollary 2.2, for all $k\ge k(L)$, $z\in Z_k$ and $T\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ we can consider a minimizing function ${\tilde{u}}^T_{k,z}$ for the minimum (2.14). Moreover, since $p>1$, for all $u\in H^1_0 \left( {1\over k}\, C_{z}\right)$ there exists the maximum

$$\begin{aligned} M(u)=\max \left\{ E\left( {\tilde{u}}^T_{k,z}+t(u-\tilde{u}^T_{k,z})\right) \ :\ t\ge 0\right\} \end{aligned}$$

(2.22)

and $M(u)\ge E^T_{k,z}$ when $u\not \equiv {\tilde{u}}^T_{k,z}$ in $T\left( {1\over k}\, C_{z}\right)$.

Lemma 2.3

For all $k\ge k(L)$, $z\in Z_k$ and $T\in \mathcal{C}_L(P_k,{\overline{\Omega }})$, there exists a function $u^T_{k,z}$ in $H^1_0 \left( T\left( {1\over k}\, C_{z}\right) \right)$ 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 $T\left( {1\over k}\, C_{z}\right)$ and

$$\begin{aligned} E(u^T_{k,z})=M(u^T_{k,z})=\min \left\{ M(u)\ :\ u\in H^1_0\left( T\left( {1\over k}\, C_{z}\right) \right) ,\ u\not \equiv {\tilde{u}}^T_{k,z},\right. \nonumber \\ \left. {} \sigma (z)[u-{\tilde{u}}^T_{k,z}]\ge 0\ \text{ in } \ T\left( {1\over k}\, C_{z}\right) \right\} . \end{aligned}$$

(2.23)

Moreover, we have $E(u^T_{k,z})\ge E^T_{k,z}$.

Proof

Let us consider a minimizing sequence $(u_i)_{i\in \mathbb {N}}$ for the minimum (2.23). Whitout any loss of generality, we can assume that

$$\begin{aligned} \int _{T\left( {1\over k}\, C_z\right) }|u_i-\tilde{u}^T_{k,z}|^{p+1}dx =1\qquad \forall i\in \mathbb {N}. \end{aligned}$$

(2.24)

It follows that this sequence is bounded in $H^1_0\left( T\left( {1\over k}\, C_z\right) \right)$. Therefore, since $p<{n+2\over n-2}$ when $n\ge 3$, up to a subsequence it converges weakly in $H^1_0$, in $L^{p+1}$ and a.e. to a function ${\hat{u}}\in H^1_0\left( T\left( {1\over k}\, C_z\right) \right)$.

Notice that the $L^{p+1}$ convergence and (2.24) imply

$$\begin{aligned} \int _{T\left( {1\over k}\, C_z\right) } |{\hat{u}}-\tilde{u}^T_{k,z}|^{p+1}dx=1, \end{aligned}$$

(2.25)

so ${\hat{u}}\not \equiv {\tilde{u}}^T_{k,z}$. Moreover, the a.e. convergence implies $\sigma (z)[{\hat{u}}-{\tilde{u}}^T_{k,z}]\ge 0$ in $T\left( {1\over k}\, C_z\right)$. We say that, indeed, the convergence is strong in $H^1_0\left( T\left( {1\over k}\, C_z\right) \right)$. In fact, arguing by contradiction, assume that (up to a subsequence)

$$\begin{aligned} \int _{T\left( {1\over k}\, C_z\right) } |{\nabla }{\hat{u}} |^2dx<\lim _{i\rightarrow \infty } \int _{T\left( {1\over k}\, C_z\right) } |{\nabla }u_i |^2dx. \end{aligned}$$

(2.26)

As a consequence, we obtain $M({\hat{u}})<\lim \limits _{i\rightarrow \infty }M(u_i)$ which is a contradiction because ${\hat{u}}\not \equiv {\tilde{u}}^T_{k,z}$ and $(u_i)_{i\in \mathbb {N}}$ is a minimizing sequence for (2.23) so that $M({\hat{u}})\ge \lim \limits _{i\rightarrow \infty }M(u_i)$.

Therefore, we can conclude that $u_i\rightarrow {\hat{u}}$ in $H^1_0\left( T\left( {1\over k}\, C_z\right) \right)$ and $\lim \limits _{i\rightarrow \infty }M(u_i)=M({\hat{u}})$. Since $p>1$, there exists ${\hat{t}}>0$ such that

$$\begin{aligned} E({\tilde{u}}^T_{k,z}+{\hat{t}}({\hat{u}}-{\tilde{u}}^T_{k,z}))=M({\hat{u}}). \end{aligned}$$

(2.27)

Thus, all the assertions of Lemma 2.3 hold with $u^T_{k,z} = {\tilde{u}}^T_{k,z}+{\hat{t}}({\hat{u}}-{\tilde{u}}^T_{k,z})$. $\square$

Proposition 2.4

There exists $k_1(L)\ge k(L)$ such that for all $k\ge k_1(L)$, $z\in Z_k$ and $T\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ the function $u^T_{k,z}$ is a solution of the Dirichlet problem

$$\begin{aligned} -\Delta u=|u|^{p-1}u+w\quad \mathrm{in }\ T\left( {1\over k}\, C_z\right) ,\qquad u=0\quad \mathrm{on }\ \partial T\left( {1\over k}\, C_z\right) . \end{aligned}$$

(2.28)

Proof

It is clear that the function ${\tilde{u}}^T_{k,z}$ (local minimum of the functional E) is a solution of the Dirichlet problem (2.28). In order to prove that, for k large enough, also $u^T_{k,z}$ is solution of the same problem, let us consider the function $G:T\left( {1\over k}\, C_z\right) \times \mathbb {R}\rightarrow \mathbb {R}$ such that $G(x,\cdot )\in \mathcal{C}^2(\mathbb {R})$ $\forall x\in T\left( {1\over k}\, C_z\right)$ and

$$\begin{aligned} \begin{array}{cl} G(x,t) = {|t|^{p+1}\over p+1}+w(x)\, t &{}\mathrm{if }\ \sigma (z)[t-{\tilde{u}}^T_{k,z}(x)]\ge 0, \\ {\partial ^2 G\over \partial t^2}\, (x,t)={\partial ^2 G\over \partial t^2}\, (x,{\tilde{u}}^T_{k,z}(x)) &{}\mathrm{if }\ \sigma (z)[t-{\tilde{u}}^T_{k,z}(x)]\le 0. \end{array} \end{aligned}$$

(2.29)

Moreover, let us set $g(x,t)= \frac{dG}{dt}(x,t)$.

Then, consider the functional $E_{k,z,T}:H^1_0\left( T \left( {1\over k}\, C_z\right) \right) \rightarrow \mathbb {R}$ defined by

$$\begin{aligned} E_{k,z,T}(u)={1\over 2}\int _{T\left( {1\over k}\, C_z\right) }|{\nabla }u|^2dx- \int _{T\left( {1\over k}\, C_z\right) }G(x,u)\, dx. \end{aligned}$$

(2.30)

Let us assume, for example, $\sigma (z)=1$ (in a similar way one can argue when $\sigma (z)=-1$).

One can verify by direct computation that for all $u\not \equiv {\tilde{u}}^T_{k,z}$ there exists a unique $t_u>0$ such that

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

(2.31)

if and only if $(u-{\tilde{u}}^T_{k,z})\vee 0\not \equiv 0$. In this case $E'_{k,z,T}({\tilde{u}}^T_{k,z}+t(u-{\tilde{u}}^T_{k,z})[u-\tilde{u}^T_{k,z}]$ is positive for $t\in ]0,t_u[$ and negative for $t>t_u$, so we have

$$\begin{aligned} E_{k,z,T}({\tilde{u}}^T_{k,z}+t_u(u-{\tilde{u}}^T_{k,z})) = \max \{ E_{k,z,T}({\tilde{u}}^T_{k,z}+t(u-{\tilde{u}}^T_{k,z}))\ :\ t >0\}. \end{aligned}$$

(2.32)

Moreover, we have

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

(2.33)

Taking into account that ${\tilde{u}}^T_{k,z}$ is solution of problem (2.28), we obtain by direct computation

$$\begin{aligned} E_{k,z,T}({\tilde{u}}^T_{k,z}+t(u-{\tilde{u}}^T_{k,z}))= & {} E_{k,z,T}({\tilde{u}}^T_{k,z}+t[(u-{\tilde{u}}^T_{k,z})\vee 0]) \nonumber \\&+{t^2\over 2} \int _{T\left( {1\over k}\, C_z\right) } |{\nabla }[(u-{\tilde{u}}^T_{k,z})\wedge 0] |^2dx \nonumber \\&- {t^2\over 2}\, p \int _{T\left( {1\over k}\, C_z\right) } |{\tilde{u}}^T_{k,z}|^{p-1}[(u-{\tilde{u}}^T_{k,z})\wedge 0]^2dx \qquad \forall t>0.\nonumber \\ \end{aligned}$$

(2.34)

Notice that

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

(2.35)

and, by (2.15),

$$\begin{aligned} \lim _{k\rightarrow \infty }\sup \left\{ \int _{T\left( {1\over k}\, C_z\right) }|{\tilde{u}}^T_{k,z}|^{p+1}dx \ :\ z\in Z_k,\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\right\} =0. \end{aligned}$$

(2.36)

Moreover, we have

$$\begin{aligned}&\int _{T\left( {1\over k}\, C_z\right) }|{\nabla }[(u-{\tilde{u}}^T_{k,z})\wedge 0] |^2dx \ge \lambda _k \left( \int _{T\left( {1\over k}\, C_z\right) } |(u-{\tilde{u}}^T_{k,z})\wedge 0|^{p+1}dx\right) ^{2\over p+1} \nonumber \\&\quad \forall k\in \mathbb {N},\ \forall z\in Z_k,\ \forall T\in \mathcal{C}_L(P_k,{\overline{\Omega }}) \end{aligned}$$

(2.37)

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

$$\begin{aligned}&\lambda _k=\inf \left\{ \int _{T\left( {1\over k}\, C_z\right) } |{\nabla }\psi |^2dx\ :\ z\in Z_k,\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }}),\ \right. \nonumber \\&\quad \left. \psi \in H^1_0\left( T\left( {1\over k}\, C_z\right) \right) ,\ \int _{T\left( {1\over k}\, C_z\right) }|\psi |^{p+1}dx=1\right\} . \end{aligned}$$

(2.38)

Notice that $\lim \limits _{k\rightarrow \infty }\lambda _k=\infty$ otherwise for all $i\in \mathbb {N}$ there exist $k_i\in \mathbb {N}$ $z_i\in Z_{k_i}$, $T_i\in \mathcal{C}_L(P_{k_i},{\overline{\Omega }})$, $\psi _i\in H^1_0\left( T_i\left( {1\over k_i}\, C_{z_i}\right) \right)$ (extended by the value zero outside $T_i\left( {1\over k_i}\, C_{z_i}\right)$) such that $\lim \limits _{i\rightarrow \infty } k_i=\infty$, $\int _{\Omega }|\psi _i|^{p+1}dx=1$ $\forall i\in \mathbb {N}$ and $\lim \limits _{i\rightarrow \infty }\int _\Omega |D\psi _i|^2dx<\infty$.

As a consequence, since $p<{n+2\over n-2}$ when $n\ge 3$, there exists ${\bar{\psi }}$ in $H^1_0(\Omega )$ such that (up to a subsequence) $\psi _i\rightarrow {\bar{\psi }}$ as $i\rightarrow \infty$ weakly in $H^1_0(\Omega )$, in $L^{p+1}(\Omega )$ and a.e. in $\Omega$.

Moreover, since $\lim \limits _{i\rightarrow \infty }\mathop {\mathrm{meas}}\nolimits T_i\left( {1\over k_i}\, C_{z_i}\right) =0$, the a.e. convergence implies that ${\bar{\psi }}\equiv 0$ in $\Omega$, which is in contradiction with the convergence in $L^{p+1}(\Omega )$ because $\int _{\Omega }|\psi |^{p+1}dx=1$ $\forall i\in \mathbb {N}$.

Thus, we can conclude that $\lim \limits _{k\rightarrow \infty }\lambda _k=\infty$. It follows that, for k large enough,

$$\begin{aligned} E_{k,z,T}({\tilde{u}}^T_{k,z}+t[(u-{\tilde{u}}^T_{k,z})\vee 0]) \le E_{k,z,T}({\tilde{u}}^T_{k,z}+t(u-{\tilde{u}}^T_{k,z}))\qquad \forall t>0,\ \forall u\in H^1_0\left( {1\over k}\, C_z\right) .\nonumber \\ \end{aligned}$$

(2.39)

As a consequence, if we denote by $\Gamma$ the set defined by

$$\begin{aligned} \Gamma =\left\{ u\in H^1_0\left( T \left( {1\over k}\, C_z\right) \right) \ :\ u\not \equiv {\tilde{u}}^T_{k,z},\ E_{k,z,T}'(u)[u-{\tilde{u}}^T_{k,z}]=0\right\} , \end{aligned}$$

(2.40)

we have $u^T_{k,z}\in \Gamma$ and

$$\begin{aligned} E(u^T_{k,z})= E_{k,z,T}(u^T_{k,z})=\min _\Gamma E_{k,z,T}. \end{aligned}$$

(2.41)

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

$$\begin{aligned} E'_{k,z,T}(u^T_{k,z})[\varphi ]= \mu \left\{ E''_{k,z,T}(u^T_{k,z})[u^T_{k,z}-{\tilde{u}}^T_{k,z},\varphi ] +E'_{k,z,T}(u^T_{k,z})[\varphi ]\right\} \quad \forall \varphi \in H^1_0 \left( {1\over k}\, C_z\right) .\nonumber \\ \end{aligned}$$

(2.42)

In particular, if we choose $\varphi =u^T_{k,z}-{\tilde{u}}^T_{k,z}$, we obtain $\mu =0$ because $E'_{k,z,T}(u^T_{k,z})[u^T_{k,z}-\tilde{u}^T_{k,z}]=0$ while $E''_{k,z,T}(u^T_{k,z})[u^T_{k,z}-\tilde{u}^T_{k,z},u^T_{k,z}-{\tilde{u}}^T_{k,z}]\ne 0$. Thus, $u^T_{k,z}$ is a weak solution of the Dirichlet problem

$$\begin{aligned} -\Delta u^T_{k,z}=g(x,u^T_{k,z})\quad \mathrm{in }\ T\left( {1\over k}\, C_z\right) ,\qquad u=0\quad \mathrm{on }\ \partial T\left( {1\over k}\, C_z\right) . \end{aligned}$$

(2.43)

On the other hand, since $u^T_{k,z}-{\tilde{u}}^T_{k,z}\ge 0$ in $T\left( {1\over k}\, C_z\right)$, we have

$$\begin{aligned} g(x,u^T_{k,z}(x)) =|u^T_{k,z}(x)|^{p-1}u^T_{k,z}(x)+w(x)\qquad \forall x\in T\left( {1\over k}\, C_z\right) , \end{aligned}$$

(2.44)

so $u^T_{k,z}$ is a solution of problem (2.28). $\square$

When the function $u^T_k=\sum \limits _{z\in Z_k}u^T_{k,z}$ satisfies a suitable stationarity property, then it is solution of problem (1.1) (here the function $u^T_{k,z}$ is extended by the value zero outside $T\left( {1\over k}\, C_z\right)$). In fact, the following proposition holds.

Proposition 2.5

Assume that $k\ge k_1(L)$ and $T\in \mathcal{C}_L(P_k,{\overline{\Omega }})$. Moreover, assume that the function $u^T_k=\sum \limits _{z\in Z_k}u^T_{k,z}$ satisfies the following condition: $E'(u^T_k)[v\cdot D u_k^T]=0$ for all vector field $v\in \mathcal{C}^1({\overline{\Omega }},\mathbb {R}^n)$ such that $v\cdot \nu =0$ on $\partial \Omega$ (here $\nu$ denotes the outward normal vector on $\partial \Omega$). Then, $u_k^T$ is a solution of the Dirichlet problem (1.1).

Proof

We have to prove that $E'(u^T_k)[\varphi ]=0$ $\forall \varphi \in H^1_0(\Omega )$. Taking into account Proposition 2.4, since $u^T_{k,z}$ satisfies the Dirichlet problem (2.28) for all $z\in Z_k$, we have

$$\begin{aligned} E'(u^T_k)[\varphi ]= & {} \int _\Omega [{\nabla }u^T_k\cdot {\nabla }\varphi - |u^T_k|^{p-1}u^T_k\varphi -w\varphi ]dx\nonumber \\= & {} \sum \limits _{z\in Z_k}\int _{T\left( {1\over k}\, C_z\right) } [{\nabla }u^T_k\cdot {\nabla }\varphi - |u^T_k|^{p-1}u^T_k\varphi -w\varphi ]dx \end{aligned}$$

(2.45)

$$\begin{aligned}= & {} \sum \limits _{z\in Z_k} \int _{\partial T\left( {1\over k}\, C_z\right) } \varphi \, ({\nabla }u^T_k\cdot \nu _{k,z})d\sigma , \end{aligned}$$

(2.46)

where $\nu _{k,z}$ denotes the outward normal on $\partial T\left( {1\over k}\, C_z\right)$. Thus, in order to obtain $E'(u^T_k)$ $[\varphi ]=0$, we have to prove that if $z_1,z_2\in Z_k$ and $|z_1-z_2|=1$ (that is ${T\left( {1\over k}\, C_{z_1}\right) }$ and ${T\left( {1\over k}\, C_{z_2}\right) }$ are adjacent subdomains of $\Omega$) then

$$\begin{aligned} {\nabla }u^T_{k,z_1}(x)={\nabla }u^T_{k,z_2}(x) \qquad \forall x\in \partial T\left( {1\over k}\, C_{z_1}\right) \cap \partial T\left( {1\over k}\, C_{z_2}\right) . \end{aligned}$$

(2.47)

Taking into account that $u^T_{k,z}$ satisfies problem (2.28) for all $z\in Z_k$, for all vector field $v\in \mathcal{C}^1({\overline{\Omega }},\mathbb {R}^n)$ such that $v\cdot \nu =0$ on $\partial \Omega$ we obtain

$$\begin{aligned} E'(u^T_k)[v\cdot {\nabla }u^T_k]= & {} \nonumber \\= & {} \int _{\Omega } [{\nabla }u^T_k\cdot {\nabla }(v\cdot {\nabla }u^T_k)-|u^T_k|^{p-1}u^T_k (v\cdot {\nabla }u^T_k)-w(v\cdot {\nabla }u^T_k)]dx \nonumber \\= & {} \sum \limits _{z\in Z_k} \int _{T\left( {1\over k}\, C_{z}\right) } [{\nabla }u^T_k\cdot {\nabla }(v\cdot {\nabla }u^T_k)-|u^T_k|^{p-1}u^T_k (v\cdot {\nabla }u^T_k)-w(v\cdot {\nabla }u^T_k)]dx \nonumber \\= & {} \sum \limits _{z\in Z_k} \int _{\partial T\left( {1\over k}\, C_{z}\right) } ( {\nabla }u^T_k\cdot \nu _{k,z})^2(v\cdot \nu _{k,z})d\sigma . \end{aligned}$$

(2.48)

Since $E'(u^T_k)[v\cdot {\nabla }u^T_k]=0$ $\forall v\in \mathcal{C}^1({\overline{\Omega }},\mathbb {R}^n)$ such that $v\cdot \nu =0$ on $\partial \Omega$, (2.47) follows easily. Thus, we can conclude that $u^T_k$ is a solution of problem (1.1). $\square$

In order to obtain a function $u^T_k$ which is stationary in the sense of Proposition 2.5, we can, for example, minimize $E(u^T_k)$ with respect to T for k large enough.

First notice that, since $\Omega$ is a smooth bounded domain, there exist $k'_\Omega \ge k_\Omega$ and $L'_\Omega \ge 1$ such that, for all $k\ge k'_\Omega$ and $L\ge L_\Omega '$, we have

$$\begin{aligned} \{T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\ :\ T(P_k)={\overline{\Omega }}\}\ne \emptyset . \end{aligned}$$

(2.49)

Moreover, using Ascoli-Arzelà Theorem, one can show the following lemma.

Lemma 2.6

If (2.49) holds, there exists $\widetilde{T}_k^L\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ such that $\widetilde{T}^L_k(P_k)={\overline{\Omega }}$ and

$$\begin{aligned} \sum _{z\in Z_k} E(u^{{\widetilde{T}}^L}_{k,z})=\min \left\{ \sum _{z\in Z_k} E(u^{ T}_{k,z})\ :\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }}),\ T(P_k)={\overline{\Omega }}\right\} . \end{aligned}$$

(2.50)

For all $L\ge 1$ and $T\in \mathcal{C}_L(P_k,{\overline{\Omega }})$, let us set

$$\begin{aligned} \mathcal{L}(T)=\inf \left\{ \mathcal{L}\ :\ \mathcal{L}\ge 1,\ {1\over \mathcal{L}}\, |x-y|\le |T(x)-T(y)|\le \mathcal{L}\, |x-y|\quad \forall x,y\in P_k\right\} .\nonumber \\ \end{aligned}$$

(2.51)

Using again Ascoli-Arzelà Theorem, we infer that, for all $L\ge L'_\Omega$ and $k\ge k'_\Omega$, there exists $T_k^L\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ such that $T_k^L\left( {1\over k}\, C_z\right) ={\widetilde{T}}_k^L\left( {1\over k}\, C_z\right)$ $\forall z\in Z_k$ and

$$\begin{aligned} \mathcal{L}(T_k^L)=\min \left\{ \mathcal{L}(T)\ :\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }}),\ T\left( {1\over k}\, C_z\right) ={\widetilde{T}}_k^L\left( {1\over k}\, C_z\right) \ \forall z\in Z_k\right\} .\nonumber \\ \end{aligned}$$

(2.52)

Notice that $T_k^L$ depends only on the geometrical properties of the subdomains ${\widetilde{T}}_k^L\left( {1\over k}\, C_z\right)$ with $z\in Z_k$. A large $\mathcal{L}(T_k^L)$ means that there are large differences in the sizes and in the shape of these subdomains.

We can now state the following multiplicity result.

Theorem 2.7

Let $n\ge 1$, $p>1$ and $p<{n+2\over n-2}$ when $n\ge 3$. Moreover, assume that there exists ${\bar{L}}\ge L'_\Omega$ such that

$$\begin{aligned} \limsup _{k\rightarrow \infty }\mathcal{L}(T_k^{{\bar{L}}})< {\bar{L}}. \end{aligned}$$

(2.53)

Then, problem (1.1) admits infinitely many solutions (see also Remark 3.1 concerning condition (2.53)).

Theorem 2.7 is a direct consequence of the following proposition.

Proposition 2.8

If the assumptions of Theorem 2.7 are satisfied, for all $w\in L^2(\Omega )$ there exists ${\bar{k}}\ge k_\Omega$ such that for all $k\ge {\bar{k}}$ there exists $T^{{\bar{L}}}_k\in \mathcal{C}_{\bar{L}}(P_k,{\overline{\Omega }})$ satisfying the following property: $T^{{\bar{L}}}_k(P_k)={\overline{\Omega }}$ and the function $u_k=\sum \limits _{z\in Z_k}u^{T^{{\bar{L}}}_k}_{k,z}$ is a solution of problem (1.1). Moreover, the number of nodal regions of $u_k$ tends to infinity as $k\rightarrow \infty$ and

$$\begin{aligned} \lim _{k\rightarrow \infty }\min \left\{ E\left( u^{T^{\bar{L}}_k}_{k,z}\right) \ :\ z\in Z_k\right\} =\infty . \end{aligned}$$

(2.54)

Proof

As a consequence of condition (2.53), there exist ${\bar{k}}\in \mathbb {N}$ and ${\bar{{\varepsilon }}}>0$ such that

$$\begin{aligned} D\circ T_k^{{\bar{L}}}\in \mathcal{C}_{\bar{L}}(P_k,{\overline{\Omega }})\quad \forall k\ge {\bar{k}},\ \forall D\in \mathcal{C}_{1+{\bar{{\varepsilon }}}}({\overline{\Omega }},{\overline{\Omega }}). \end{aligned}$$

(2.55)

Moreover, from Proposition 2.4 we infer that, if we choose ${\bar{k}}$ large enough, for all $k\ge {\bar{k}}$ and $z\in Z_k$ the function $u^{T^{{\bar{L}}}_k}_{k,z}$ is a solution of the Dirichlet problem

$$\begin{aligned} -\Delta u=|u|^{p-1}u+w\quad \text{ in } T_k^{\bar{L}}\left( {1\over k}\, C_z\right) ,\qquad u=0\quad \text{ on } \partial T_k^{{\bar{L}}}\left( {1\over k} \,C_z\right) . \end{aligned}$$

(2.56)

Thus, taking into account Proposition 2.5 we have to prove that $E'(u_k)[v\cdot Du_k]=0$ for all vector field $v\in \mathcal{C}^1(\Omega ,\mathbb {R}^n)$ such that $v\cdot \nu =0$ on $\partial \Omega$.

Therefore, for all vector field $v\in \mathcal{C}^1({\overline{\Omega }}, \mathbb {R}^n)$ such that $v\cdot \nu =0$ on $\partial \Omega$ and for all $\tau \in \mathbb {R}$, let us consider the function $D_\tau :{\overline{\Omega }}\rightarrow {\overline{\Omega }}$ defined by the Cauchy problem

$$\begin{aligned} {\partial D_\tau (x)\over \partial \tau }=v\circ D_\tau (x),\qquad D_0(x)=x\qquad \forall \tau \in \mathbb {R},\ \forall x\in {\overline{\Omega }}. \end{aligned}$$

(2.57)

Then, we have $D_\tau ({\overline{\Omega }})={\overline{\Omega }}$ $\forall \tau \in \mathbb {R}$ and

$$\begin{aligned} \lim _{\tau \rightarrow 0}\mathcal{L}(D_\tau \circ T^{{\bar{L}}}_k)=\mathcal{L}(T^{{\bar{L}}}_k), \end{aligned}$$

(2.58)

so there exists ${\bar{\tau }}>0$ such that $D_\tau \circ T^{{\bar{L}}}_k\in \mathcal{C}_{{\bar{L}}}(P_k,{\overline{\Omega }})$ $\forall \tau \in [-{\bar{\tau }},{\bar{\tau }}]$. It follows that

$$\begin{aligned} E(u_{k})=\sum _{z\in Z_k}E(u^{T_k^{{\bar{L}}}}_{k,z}) \le \sum _{z\in Z_k}E(u^{D_\tau \circ T_k^{{\bar{L}}} }_{k,z}) = E(u^{D_\tau \circ T_k^{{\bar{L}}} }_{k})\quad \forall \tau \in [-{\bar{\tau }},{\bar{\tau }}]. \end{aligned}$$

(2.59)

Notice that

$$\begin{aligned} {d\over d\tau }\, E(u_k\circ D^{-1}_\tau )_{|_{\tau =0}}=-E'(u_k)[v\cdot {\nabla }u_k]. \end{aligned}$$

(2.60)

Thus, we have to prove that

$$\begin{aligned} {d\over d\tau }\, E(u_k\circ D^{-1}_\tau )_{|_{\tau =0}}=0. \end{aligned}$$

(2.61)

For the proof, we argue by contradiction and assume that (2.61) does not hold. For example, we assume that

$$\begin{aligned} {d\over d\tau }\, E(u_k\circ D^{-1}_\tau )_{|_{\tau =0}}<0 \end{aligned}$$

(2.62)

(otherwise we replace v by $-v$). As a consequence, there exists a sequence of positive numbers $(\tau _i)_{i\in \mathbb {N}}$ such that $\lim \limits _{i\rightarrow \infty }\tau _i=0$ and $E(u_k\circ D^{-1}_{\tau _i})<E(u_k)$ $\forall i\in \mathbb {N}$. From Corollary 2.2 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 ${\tilde{u}}^{D_{\tau _i}\circ T^{{\bar{L}}}_k}_{k,z}$ and $\tilde{u}^{D_{\tau _i}\circ T^{{\bar{L}}}_k}_{k,z}\rightarrow {\tilde{u}}_{k,z}^{T^{\bar{L}}_k}$ as $i\rightarrow \infty$ in $H^1_0(\Omega )$ $\forall k\ge {\bar{k}}$, $\forall z\in Z_k$.

As in the proof of Proposition 2.4, let us consider the functions $G^i_{k,z}$ verifying

$$\begin{aligned} \begin{array}{ll}G^i_{k,z}(x,t) = {|t|^{p+1}\over p+1}+w(x)\, t &{}\mathrm{if }\ \sigma (z)[t-\tilde{u}^{D_{\tau _i}\circ T^{{\bar{L}}}_k}_{k,z}(x)]\ge 0, \\ {\partial ^2 G^i_{k,z}\over \partial t^2}\, (x,t)={\partial ^2 G^i_{k,z}\over \partial t^2}\, (x,{\tilde{u}}^{D_{\tau _i}\circ T^{\bar{L}}_k}_{k,z}(x)) &{}\mathrm{otherwise}, \end{array} \end{aligned}$$

(2.63)

the functional $E^i_{k,z}:H^1_0\left( T^{{\bar{L}}}_k\left( {1\over k}\, C_z\right) \right) \rightarrow \mathbb {R}$ defined by

$$\begin{aligned} E_{k,z,i}(u)={1\over 2}\int _{T^{{\bar{L}}}_k\left( {1\over k}\, C_z\right) }|{\nabla }u|^2dx-\int _{T^{{\bar{L}}}_k\left( {1\over k}\, C_z\right) }G^i_{k,z}(x,u)\, dx \end{aligned}$$

(2.64)

and the manifold

$$\begin{aligned} \Gamma ^i_{k,z} =\left\{ u\in H^1_0\left( T^{{\bar{L}}}_k\left( {1\over k}\, C_z\right) \right) \ :\ u\not \equiv {\tilde{u}}^{D_{\tau _i}\circ T^{{\bar{L}}}_k}_{k,z},\ E_{k,z,i}'(u)[u-{\tilde{u}}^{D_{\tau _i}\circ T^{{\bar{L}}}_k}_{k,z}]=0\right\} .\nonumber \\ \end{aligned}$$

(2.65)

We say that

$$\begin{aligned} \max \left\{ \sum _{z\in Z_k}E\left( {\tilde{u}}^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_{\tau _i}+t_z \left( u^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_{\tau _i}-{\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}\right) \right) : t_z\ge 0,\ \forall z\in Z_k\right\} <E(u_k)\nonumber \\ \end{aligned}$$

(2.66)

for i large enough.

In fact, arguing by contradiction, assume that (up to a subsequence still denoted by $(\tau _i)_{i\in \mathbb {N}}$) the inequality (2.66) does not hold. Then, for all $i\in \mathbb {N}$ and $z\in Z_k$, there exists $t_{z,i}\ge 0$ such that

$$\begin{aligned} \sum _{z\in Z_k}E\left( {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}+t_{z,i} \left( u^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}- {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}\right) \right) \ge E(u_k)\qquad \forall i\in \mathbb {N}.\qquad \end{aligned}$$

(2.67)

It follows that $\lim \limits _{i\rightarrow \infty }t_{z,i}=1$ $\forall z\in Z_k$ and

$$\begin{aligned} \sum _{z\in Z_k}E\left( {\tilde{u}}^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_{\tau _i}+t_{z,i} \left( u^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_{\tau _i}- {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}\right) \right) \end{aligned}$$

$$\begin{aligned} \ge \sum _{z\in Z_k}E\left( {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}+ t_{z,i} \left( u^{T^{{\bar{L}}}_k}_{k,z}- {\tilde{u}}^{T^{\bar{L}}_k}_{k,z}\right) \right) \qquad \forall i\in \mathbb {N}\end{aligned}$$

(2.68)

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

$$\begin{aligned} {d\over d\tau }\, E\left( u^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_\tau \right) _{|_{\tau =0}}\ge 0 \end{aligned}$$

(2.69)

in contradiction with (2.62). Thus, (2.66) holds.

Notice that, if ${\bar{k}}$ is chosen large enough,

$$\begin{aligned}&E'\left( {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}+t \left( u^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}- {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}\right) \right) \cdot \nonumber \\&\quad \cdot \left[ {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i} -{\tilde{u}}^{D_{\tau _i}\circ T^{{\bar{L}}}_k}_{k,z} +t \left( u^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}- {\tilde{u}}^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_{\tau _i}\right) \right] , \end{aligned}$$

(2.70)

for i large enough, is positive for $t=\Vert u^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_{\tau _i}- {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}\Vert ^{-1}_{H^1_0}$ and tends to $-\infty$ as $t\rightarrow \infty$. As a consequence, there exists $t^i_{k,z}\in \mathbb {R}$ such that

$$\begin{aligned} {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}+t^i_{k,z} \left( u^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}- {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}\right) \in \Gamma ^i_{k,z}. \end{aligned}$$

(2.71)

Therefore, from (2.66) and (2.71) we obtain

$$\begin{aligned} E(u_k)> & {} \max \left\{ \sum _{z\in Z_k}E\left( {\tilde{u}}^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}+t_z \left( u^{T^{{\bar{L}}}_k}_{k,z}\circ D^{-1}_{\tau _i}- {\tilde{u}}^{T^{\bar{L}}_k}_{k,z}\circ D^{-1}_{\tau _i}\right) \right) : t_z\ge 0\ \forall z\in Z_k\right\} \nonumber \\\ge & {} \sum _{z\in Z_k}E\left( u^{D_{\tau _i}\circ T_k^{{\bar{L}}}}_{k,z}\right) = E(u^{D_{\tau _i}\circ T_k^{{\bar{L}}}}_{k} ) \end{aligned}$$

(2.72)

for i large enough, in contradiction with (2.59).

Thus, we can conclude that ${d\over d\tau }E(u_k\circ D^{-1}_\tau )_{|_{\tau =0}}=0$ that is $E'(u_k)[v\cdot Du_k]=0$ for all vector field $v\in \mathcal{C}^1({\overline{\Omega }},\mathbb {R}^n)$ such that $v\cdot \nu =0$ on $\partial \Omega$, so $u_k$ is a solution of problem (1.1).

Notice that, if J(k) denotes the number of elements of $Z_k$, the solution $u_k$ has at least J(k) nodal regions for k large enough. Moreover, we have

$$\begin{aligned} \lim _{k\rightarrow \infty } {J(k)\over k^n}=\mathop {\mathrm{meas}}\nolimits (\Omega ), \end{aligned}$$

(2.73)

so the number of nodal regions of $u_k$ tends to infinity as $k\rightarrow \infty$.

Finally, notice that (2.54) follows directly from Lemma 2.1 and Lemma 2.3.

So the proof is complete. $\square$

Let us point out that, if $n=1$, condition (2.53) in Theorem 2.7 is satisfied. In fact, it is a consequence of the following lemma (see also Remark 3.1 concerning the case $n>1$).

Lemma 2.9

Assume $n=1$, $p>1$ and $w\in L^2(\Omega )$. Then, for all $L>1$ there exists ${\bar{k}}(L)\in \mathbb {N}$ such that

$$\begin{aligned} \{T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\ :\ T(P_k)={\overline{\Omega }}\}\ne \emptyset \qquad \forall k\ge {\bar{k}}(L). \end{aligned}$$

(2.74)

Moreover,

$$\begin{aligned} \lim _{k\rightarrow \infty } \mathcal{L}(T^L_k)=1\qquad \forall L>1. \end{aligned}$$

(2.75)

Proof

Let $\Omega =]a,b[$. First notice that, if $Z_k$ consists of J(k) points $z_1,\ldots ,z_{J(k)}$, then ${J(k)\over k}\le b-a$ and $\lim \limits _{k\rightarrow \infty }{J(k)\over k}=b-a$. Since $L>1$, it follows that there exists ${\bar{k}}(L)\in \mathbb {N}$ such that ${J(k)\over k}>{b-a\over L}$ $\forall k\ge {\bar{k}}(L)$, which implies (2.74) as one can easily verify. Also, notice that in this case we have

$$\begin{aligned}&\mathcal{L}(T^L_k)=\min \left\{ \mathcal{L}\ :\ \mathcal{L}\ge 1,\ {1\over \mathcal{L}}\le k\, \mathop {\mathrm{meas}}\nolimits \left[ T^L_k\left( {1\over k}\, C_{z_j}\right) \right] \le \mathcal{L}\right. \nonumber \\&\quad \left. \text{ for } j=1,\ldots ,J(k)\right\} \qquad \forall k > {\bar{k}}(L). \end{aligned}$$

(2.76)

Moreover, if we denote by ${\bar{u}}^{T^L_k}_{k,z}$ the function $u^{T^L_k}_{k,z}$ obtained when $w=0$, we get by direct computation

$$\begin{aligned} \limsup _{k\rightarrow \infty } {1\over k^{p+3\over p-1}}\, \max \left\{ E(\bar{u}^{T^L_k}_{k,z})\ :\ z\in Z_k\right\} <\infty \end{aligned}$$

(2.77)

and

$$\begin{aligned} \liminf _{k\rightarrow \infty } {1\over k^{p+3\over p-1}}\, \min \left\{ E(\bar{u}^{T^L_k}_{k,z})\ :\ z\in Z_k\right\} >0. \end{aligned}$$

(2.78)

Taking into account that

$$\begin{aligned}&\left| \int _{T^L_k\left( {1\over k}\, C_z\right) }u\, w\, dx\right| \le \left( \int _{T^L_k\left( {1\over k}\, C_z\right) }u^2\, dx\right) ^{1\over 2}\Vert w\Vert _{L^2(\Omega )} \\&\quad \le \left[ \mathop {\mathrm{meas}}\nolimits T^L_k\left( {1\over k}\, C_{z}\right) \right] ^{{1\over 2}-{1\over p+1}}\Vert u\Vert _{L^{p+1}\left( T^L_k\left( {1\over k}\, C_z\right) \right) }\Vert w\Vert _{L^2(\Omega )}\nonumber \\&\quad \forall z\in Z_k,\nonumber \end{aligned}$$

(2.79)

for all $w\in L^2(\Omega )$ we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }{1\over k^{5-p\over 2(p-1)}}\max \{|E( u^{T^L_k}_{k,z})- E({\bar{u}}^{T^L_k}_{k,z})|\ :\ z\in Z_k\}<\infty \end{aligned}$$

(2.80)

and, as a consequence,

$$\begin{aligned} \lim _{k\rightarrow \infty } {J(k)\over k^{p+3\over 2(p-1)}} \max \{|E( u^{T^L_k}_{k,z})- E({\bar{u}}^{T^L_k}_{k,z})|\ :\ z\in Z_k\}<\infty . \end{aligned}$$

(2.81)

It is clear that $1\le \mathcal{L}(T^L_k)\le L$, so $\liminf \limits _{k\rightarrow \infty }\mathcal{L}(T^L_k)\ge 1$. Arguing by contradiction, assume that (2.75) does not hold, that is

$$\begin{aligned} \limsup _{k\rightarrow \infty }\mathcal{L}(T^L_{k})>1. \end{aligned}$$

(2.82)

Thus, there exists a sequence $(k_i)_{i\in \mathbb {N}}$ in $\mathbb {N}$ such that

$$\begin{aligned} \lim _{i\rightarrow \infty }\mathcal{L}(T^L_{k_i})>1. \end{aligned}$$

(2.83)

Taking into account (2.76), it follows that there exists a sequence $(z'_i)_{i\in \mathbb {N}}$ such that $z'_i\in Z_{k_i}$ $\forall i\in \mathbb {N}$ and (up to a subsequence)

$$\begin{aligned} \lim _{i\rightarrow \infty }k_i\mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{z'_i}\right) \right] >1, \end{aligned}$$

(2.84)

or there exists a sequence $(z_i'')_{i\in \mathbb {N}}$ such that $z_i''\in Z_{k_i}$ $\forall i\in \mathbb {N}$ and (up to a subsequence)

$$\begin{aligned} \lim _{i\rightarrow \infty }k_i\mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{z_i''}\right) \right] <1. \end{aligned}$$

(2.85)

Let us consider first the case where (2.84) holds. In this case, for all $i\in \mathbb {N}$, choose ${\tilde{z}}_i'$ in $Z_{k_i}$ such that

$$\begin{aligned} \mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{\tilde{z}_i'}\right) \right] = \min \left\{ \mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{z}\right) \right] \ : \ z\in Z_{k_i}\right\} . \end{aligned}$$

(2.86)

Then, taking into account that

$$\begin{aligned} (b-a)=\sum _{z\in Z_{k_i}}\mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{z}\right) \right] \ge J(k_i)\mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{{\tilde{z}}_i'}\right) \right] \end{aligned}$$

(2.87)

and that $\lim \limits _{i\rightarrow \infty }{J(k_i)\over k_i}=b-a$, we obtain

$$\begin{aligned} \limsup _{i\rightarrow \infty }k_i\cdot \mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{\tilde{z}_i'}\right) \right] \le 1. \end{aligned}$$

(2.88)

As one can easily verify, for all $i\in \mathbb {N}$, there exists a function $T'_i\in \mathcal{C}_L(P_{k_i},{\overline{\Omega }})$ such that $T'_i(P_{k_i})={\overline{\Omega }}$,

$$\begin{aligned}&\mathop {\mathrm{meas}}\nolimits \left[ T'_i\left( {1\over k_i}\, C_{ z}\right) \right] =\mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{z}\right) \right] \qquad \forall z\in Z_{k_i}\setminus \{z'_i,\tilde{z}'_i\}, \end{aligned}$$

(2.89)

$$\begin{aligned}&\mathop {\mathrm{meas}}\nolimits \left[ T'_i\left( {1\over k_i}\, C_{z_i'}\right) \right] = \mathop {\mathrm{meas}}\nolimits \left[ T'_i\left( {1\over k_i}\, C_{{\tilde{z}}_i'}\right) \right] \end{aligned}$$

(2.90)

and

$$\begin{aligned} \lim _{i\rightarrow \infty }\max \{|T'_i(x)-T^L_{k_i}(x)|\ :\ x\in P_{k_i}\}=0. \end{aligned}$$

(2.91)

Taking into account that $E({\bar{u}}^{T_i'}_{k_i,z})=E(\bar{u}^{T_{k_i}^L}_{k_i,z})$ $\forall z\in Z_{k_i}\setminus \{z_i',\tilde{z}_i'\}$, from (2.81) we infer that

$$\begin{aligned} \lim _{i\rightarrow \infty }{1\over k_i^{p+3\over p-1}}\sum _{z\in Z_{k_i}\setminus \{ z_i',{\tilde{z}}_i'\}}|E(u^{T_i'}_{k_i,z}) -E( u^{T_{k_i}^L}_{k_i,z}) |=0. \end{aligned}$$

(2.92)

On the other hand,

$$\begin{aligned} \liminf _{i\rightarrow \infty }{1\over k_i^{p+3\over p-1}}[E(u^{T_i'}_{k_i,z_i'}) +E( u^{T_i'}_{k_i,{\tilde{z}}_i'})]>0 \end{aligned}$$

(2.93)

as one can verify by direct computation. Moreover, (2.84) and (2.88) imply

$$\begin{aligned} \liminf _{i\rightarrow \infty } {E(u^{T^L_{k_i}}_{k_i,z_i'}) +E( u^{T^L_{k_i}}_{k_i,{\tilde{z}}_i'}) \over E(u^{T_i'}_{k_i,z_i'}) +E( u^{T_i'}_{k_i,{\tilde{z}}_i'})} >1. \end{aligned}$$

(2.94)

It follows that

$$\begin{aligned} \liminf _{i\rightarrow \infty }{1\over k_i^{p+3\over p-1}} \sum _{z\in Z_{k_i}} [E(u^{T^L_{k_i}}_{k_i,z}) -E( u^{T'_{i}}_{k_i, z})]>0, \end{aligned}$$

(2.95)

which is a contradiction because

$$\begin{aligned} \sum _{z\in Z_{k_i}} E(u^{T^L_{k_i}}_{k_i,z}) \le \sum _{z\in Z_{k_i}} E( u^{T'_{i}}_{k_i, z}) \qquad \forall i\in \mathbb {N}. \end{aligned}$$

(2.96)

When the case (2.85) occurs, we argue in analogous way. In this case, for all $i\in \mathbb {N}$ we choose ${\tilde{z}}''$ in $Z_{k_i}$ such that

$$\begin{aligned} \mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{\tilde{z}_i''}\right) \right] = \max \left\{ \mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{z}\right) \right] \ : \ z\in Z_{k_i}\right\} , \end{aligned}$$

(2.97)

which implies

$$\begin{aligned} \liminf _{i\rightarrow \infty }k_i\cdot \mathop {\mathrm{meas}}\nolimits \left[ T^L_{k_i}\left( {1\over k_i}\, C_{\tilde{z}_i''}\right) \right] \ge 1. \end{aligned}$$

(2.98)

Moreover, we can consider a function $T''_i\in \mathcal{C}_L(P_{k_i},{\overline{\Omega }})$ satisfying all the properties of $T'_i$ with $z_i''$ and ${\tilde{z}}_i''$ instead of $z_i'$ and ${\tilde{z}}_i'$.

Then, we can repeat for $T_i''$, $z_i''$ and ${\tilde{z}}_i''$ the same arguments as before. In particular, the property

$$\begin{aligned} \liminf _{i\rightarrow \infty } {E(u^{T^L_{k_i}}_{k_i,z_i''}) +E( u^{T^L_{k_i}}_{k_i,{\tilde{z}}_i''}) \over E(u^{T_i''}_{k_i,z_i''}) +E( u^{T_i''}_{k_i,{\tilde{z}}_i''})} >1 \end{aligned}$$

(2.99)

(analogous to (2.94)) now follows from (2.85) and (2.98). Thus, also in this case we obtain again a contradiction with the minimality property of $\sum \limits _{z\in Z_{k_i}}E(u^{T^L_{k_i}}_{k_i,z})$. So the proof is complete. $\square$

As a direct consequence of Theorem 2.7, Proposition 2.8 and Lemma 2.9 we obtain the following corollary.

Corollary 2.10

Assume $n=1$ and $p>1$. Then, for all $w\in L^2(\Omega )$, problem (1.1) has infinitely many solutions. More precisely, for all $L>1$ there exists ${\bar{k}}(L)\ge k_\Omega$ such that for all $k\ge {\bar{k}}(L)$ there exists $T^L_k\in \mathcal{C}_L(P_k,{\overline{\Omega }})$ such that $T^L_k(P_k)={\overline{\Omega }}$ and the function $u_k=\sum \limits _{z\in Z_k} u^{T^L_k}_{k,z}$ is a solution of problem (1.1).

Moreover, the number of nodal regions of $u_k$ tends to infinity as $k\rightarrow \infty$ and

$$\begin{aligned} \lim _{k\rightarrow \infty }\mathcal{L}(T^L_k)=1,\quad \lim _{k\rightarrow \infty }\min \{E(u^{T^L_k}_{k,z})\ :\ z\in Z_k\}=\infty \quad \forall L>1. \end{aligned}$$

(2.100)

3 Final remarks

Notice that the method we used in Sect. 2 to find infinitely many solutions of problem (1.1) with a large number of nodal regions having a prescribed structure (a check structure) may be used also in other elliptic problems as we show in this section.

It is clear that in this method condition (2.53) plays a crucial role. In Sect. 2 this condition is proved only in the case $n=1$. In next remark, we discuss about the case $n>1$.

Remark 3.1

Assume that condition (2.53) does not hold. Then, there exists a sequence $(L_i)_{i\in \mathbb {N}}$ in $\mathbb {R}$ such that

$$\begin{aligned} \lim \limits _{i\rightarrow \infty }L_i=\infty \qquad \mathrm{and }\qquad \limsup _{k\rightarrow \infty }\mathcal{L}(T^{L_i}_{k})=L_i\quad \forall i\in \mathbb {N}. \end{aligned}$$

(3.1)

As a consequence, we can construct a sequence $(k_i)_{i\in \mathbb {N}}$ such that

$$\begin{aligned} \lim _{i\rightarrow \infty }k_i=\infty ,\qquad \lim _{i\rightarrow \infty }\min \big \{E\big (u^{T^{L_i}_{k_i}}_{k_i,z}\big )\ :\ z\in Z_{k_i}\big \}=\infty ,\qquad \lim _{i\rightarrow \infty }\mathcal{L}(T^{L_i}_{k_i})=\infty . \end{aligned}$$

(3.2)

Notice that $\mathcal{L}(T^{L_i}_{k_i})$ is large, for example, when there are large differences in the sizes or in the shapes of the subdomains $T^{L_i}_{k_i}\left( {1\over k_i}\, C_z\right)$ with $z\in Z_{k_i}$. For $k_i$ large enough, too large differences seem to be incompatible with the minimality property

$$\begin{aligned} \sum _{z\in Z_{k_i}}E\big (u^{T^{L_i}_{k_i}}_{k_i,z}\big )=\min \left\{ \sum _{z\in Z_{k_i}}E\big (u^T_{k_i,z}\big )\ :\ T\in \mathcal{C}_{L_i}(P_{k_i},{\overline{\Omega }}),\ T(P_{k_i})={\overline{\Omega }}\right\} \quad \forall i\in \mathbb {N}.\nonumber \\ \end{aligned}$$

(3.3)

This fact explains why condition (2.53) holds in the case $n=1$. In the case $n>1$, on the contrary, even if the subdomains $T^{L_i}_{k_i}\left( \frac{1}{k_i}C_z\right)$ with $z\in Z_{k_i}$ have all the same shape and the same size, we cannot exclude that $\mathcal{L}\left( T^{L_i}_{k_i}\right)$ is large as a consequence of the fact that the shape of these subdomains is very different from the cubes of $\mathbb {R}^n$. This explains why it is difficult to prove that condition (2.53) holds also for $n>1$.

Therefore, in the case $n>1$, the natural idea is to restrict the class of the admissible deformations of the nodal regions.

For example, we can fix ${\tilde{L}}\ge L'_\Omega$, $T_0\in \mathcal{C}_{\tilde{L}}({\overline{\Omega }},{\overline{\Omega }})$, $r>0$ and consider the set of deformations

$$\begin{aligned} \Theta ^{{\tilde{L}}}_k(T_0,r)=\{T\in \mathcal{C}_{\tilde{L}}(P_k,{\overline{\Omega }})\ :\ T(P_k)={\overline{\Omega }},\ d_k(T,T_0)\le r\} \end{aligned}$$

(3.4)

where

$$\begin{aligned} d_k(T,T_0)=\sup _{P_k}|T-T_0|+\mathop {\mathrm{Lip}}\nolimits (T-T_0) \end{aligned}$$

(3.5)

with

$$\begin{aligned} \mathop {\mathrm{Lip}}\nolimits (T-T_0)=\sup \{|x-y|^{-1}|T(x)-T_0(x)-T(y)+T_0(y)|\ :\ x,y \text { in }P_k,\ x\ne y\}.\nonumber \\ \end{aligned}$$

(3.6)

Then, arguing exactly as in Sect. 2 but minimizing in the subset $\Theta ^{{\tilde{L}}}_k(T_0,r)$ (instead of the set (2.49)), we obtain a minimizing deformation $T^{{\tilde{L}},r}_k$ which, for k large enough, gives rise to a solution $u^{{\tilde{L}},r}_k$ of problem (1.1) provided the condition

$$\begin{aligned} \limsup _{k\rightarrow \infty }\mathcal{L}(T^{{\tilde{L}},r}_k)<{\tilde{L}}, \quad \limsup _{k\rightarrow \infty }d_k(T^{{\tilde{L}},r}_k, T_0)<r \end{aligned}$$

(3.7)

(analogous to condition (2.53)) is satisfied.

It is clear that condition (3.7) holds or fails depending on the choice of ${\tilde{L}},T_0$ and r that have to be chosen in a suitable way. For example, in the case $n=1$, if we choose $T_0(x)=x$ $\forall x\in \Omega$, (3.7) holds for all $\tilde{L}>1$ and $r>0$ as follows from Lemma 2.9.

In the case $n>1$, condition (3.7) seems to have more chances than condition (2.53) to be satisfied. In fact, as we show in a paper in preparation, a variant of this method works for example when $\Omega$ is a cube of $\mathbb {R}^n$ with $n>1$, $p>1$, $p<\frac{n+2}{n-2}$ if $n>2$ and, for all $w\in L^2(\Omega )$, allows us to find infinitely many solutions $u_k(x)$ such that the nodal regions of $u_k\left( \frac{x}{k}\right)$, after translations, tend to the cube as $k\rightarrow \infty$.

Therefore, it seems quite natural to expect that, by a suitable choice of ${\tilde{L}}$, $T_0$ and r, for every bounded domain $\Omega$ in $\mathbb {R}^n$ with $n>1$ and for all $w\in L^2(\Omega )$ one can find infinitely many nodal solutions of problem (1.1) with $p>1$ and $p<\frac{n+2}{n-2}$ if $n>2$. $\square$

Notice that this method to construct solutions with nodal regions having this check structure works for more general nonlinearities, even when they are not perturbations of symmetric nonlinearities: for example when in problem (1.1) the term $|u|^{p-1}u+w$ is replaced by $c_+(u^+)^p-c_-(u^-)^p+w$ with $c_+>0$ and $c_->0$.

In fact, this method does not require any technique of deformation from the symmetry. For example, let us show how Lemma 2.9 has to be modified in this case.

In this case the energy functional is

$$\begin{aligned} F(u)={1\over 2}\int _\Omega |{\nabla }u|^2dx-{c_+\over p+1}\int _\Omega (u^+)^{p+1}dx-{c_-\over p+1}\int _\Omega (u^-)^{p+1}dx-\int _\Omega wu\, dx.\nonumber \\ \end{aligned}$$

(3.8)

We denote by $F_0$ the functional F when $w=0$.

Now, consider the number ${\hat{L}}\ge 1$ defined by ${\hat{L}}={1\over \min \{{\hat{t}},2-{\hat{t}}\}}$ where ${\hat{t}}\in ]0,2[$ is the unique number such that

$$\begin{aligned}&\min \{F_0(u)\ :\ u\in H^1_0(]0,2[),\ u\ge 0\ \mathrm{in }\ ]0,{\hat{t}}[,\ u\le 0\ \mathrm{in }\ ]{\hat{t}},2[, \nonumber \\&\quad u^+\not \equiv 0,\ u^-\not \equiv 0,\ F'_0(u)[u^+]=0,\ F'_0(u)[u^-]=0\} \\&\quad =\min \{F_0(u)\ :\ u\in H^1_0(]0,2[),\ u^+\not \equiv 0,\ u^-\not \equiv 0,\ F'_0(u)[u^+]=0,\ F'_0(u)[u^-]=0\}.\nonumber \end{aligned}$$

(3.9)

Notice that ${\hat{t}}=1$ (and so ${\hat{L}}=1$) if and only if $c_+=c_-$.

Then, we have the following lemma which extends Lemma 2.9.

Lemma 3.2

Assume $n=1$, $c_+>0$, $c_->0$, $p>1$. Then, for all $L>{\hat{L}}$ there exists ${\hat{k}}(L)\in \mathbb {N}$ such that $\{T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\ :\ T(P_k)={\overline{\Omega }}\}\ne \emptyset$ $\forall k\ge {\hat{k}}(L)$ and $\lim \limits _{k\rightarrow \infty }\mathcal{L}(T^L_k)={\hat{L}}$.

Proof

Here we describe only how the proof of Lemma 2.9 has to be modified in order to be adapted in this case.

First notice that, since $L>{\hat{L}}$ and ${\hat{L}}\ge 1$, there exists ${\hat{k}}(L)\in \mathbb {N}$ such that

$$\begin{aligned} \{T\in \mathcal{C}_L(P_k,{\overline{\Omega }})\ :\ T(P_k)={\overline{\Omega }}\}\ne \emptyset \qquad \forall k\ge {\hat{k}}(L) \end{aligned}$$

(3.10)

as one can verify as in the proof of Lemma 2.9.

In order to prove that $\lim \limits _{k\rightarrow \infty }\mathcal{L}(T^L_k)={\hat{L}}$, we argue by contradiction and assume that there exists a sequence $(k_i)_{i\in \mathbb {N}}$ in $\mathbb {N}$ such that $\lim \limits _{i\rightarrow \infty } k_i=\infty$ and $\lim \limits _{i\rightarrow \infty }\mathcal{L}(T^L_{k_i})\ne {\hat{L}}$.

First, notice that the case

$$\begin{aligned} \lim _{i\rightarrow \infty }\mathcal{L}(T^L_{k_i})<{\hat{L}} \end{aligned}$$

(3.11)

cannot happen. In fact, for all $i\in \mathbb {N}$ we can choose ${\hat{z}}_i$ and ${\hat{z}}_i+1$ in $Z_{k_i}$ such that (up to a subsequence)

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\left[ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i}\right) +\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i+1}\right) \right] \le 2. \end{aligned}$$

(3.12)

Taking into account the minimality property

$$\begin{aligned} \sum _{z\in Z_{k_i}}F(u^{T^L_{k_i}}_{k_i,z})=\min \left\{ \sum _{z\in Z_{k_i}}F(u^T_{k_i,z})\ :\ T\in \mathcal{C}_L(P_k,{\overline{\Omega }}),\ T(P_{k_i})={\overline{\Omega }}\right\} \qquad \forall i\in \mathbb {N},\nonumber \\ \end{aligned}$$

(3.13)

it follows that

$$\begin{aligned} \lim _{i\rightarrow \infty } \frac{2\min \{\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i}\right) ,\, \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i+1}\right) \}}{ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i}\right) +\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i+1}\right) }=\min \{{\hat{t}},2-{\hat{t}}\}. \end{aligned}$$

(3.14)

As a consequence, we obtain

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\min \left\{ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i}\right) ,\, \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{{\hat{z}}_i+1}\right) \right\} \le \min \{{\hat{t}},2-{\hat{t}}\} \end{aligned}$$

(3.15)

which implies

$$\begin{aligned} \lim _{i\rightarrow \infty }\mathcal{L}(T^L_{k_i})\ge {1\over \min \{{\hat{t}},2-{\hat{t}}\}}={\hat{L}}. \end{aligned}$$

(3.16)

In order to prove that $\lim _{i\rightarrow \infty }\mathcal{L}(T^L_{k_i})={\hat{L}}$, arguing by contradiction, assume that$\lim _{i\rightarrow \infty }\mathcal{L}(T^L_{k_i})$ $>{\hat{L}}$.

As a consequence, since

$$\begin{aligned} \mathcal{L}(T^L_{k_i})=\min \left\{ \mathcal{L}\ :\ \mathcal{L}\ge 1, \ {1\over \mathcal{L}}\le k_i\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_z\right) \le \mathcal{L}\ \ \forall z\in Z_{k_i}\right\} , \end{aligned}$$

(3.17)

there exists a sequence $(z'_i)_{i\in \mathbb {N}}$ such that $z'_i\in Z_{k_i}$ $\forall i\in \mathbb {N}$ and

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{z'_i}\right) >{\hat{L}}, \end{aligned}$$

(3.18)

or there exists a sequence $(z''_i)_{i\in \mathbb {N}}$ such that $z''_i\in Z_{k_i}$ $\forall i\in \mathbb {N}$ and

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{z''_i}\right) <{1\over {\hat{L}}}. \end{aligned}$$

(3.19)

Assume, for example, that ${\hat{t}}\le 1$ (otherwise we argue in a similar way but with ${\hat{t}}$ replaced by $2-{\hat{t}}$). Then, ${\hat{L}}={1\over {\hat{t}}}$ and, if ${\hat{t}}=1$, Lemma 2.9 applies. Thus, it remains to consider the case ${\hat{t}}\in ]0,1[$.

Consider first the case where (3.18) holds. Notice that there exists a sequence $(\zeta '_i)_{i\in \mathbb {N}}$ such that $\zeta '_i\in Z_{k_i}$ and $|z_i'-\zeta '_i|=1$ $\forall i \in \mathbb {N}$.

Then, the minimality property (3.13) implies

$$\begin{aligned} \lim _{i\rightarrow \infty } \frac{2\max \{\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z'_i}\right) ,\, \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta '_{i}}\right) \}}{ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z'_i}\right) +\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta '_{i}}\right) }= 2-{\hat{t}} \end{aligned}$$

(3.20)

and, arguing as in the proof of Lemma 2.9 but with $\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z'_i}\right) +\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta '_{i}}\right)$ instead of $\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z'_i}\right)$ also

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\left[ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z'_i}\right) +\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta '_{i}}\right) \right] = 2. \end{aligned}$$

(3.21)

As a consequence of (3.20) and (3.21), we obtain

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\max \left\{ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z'_i}\right) ,\, \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta '_{i}}\right) \right\} = 2-{\hat{t}} \end{aligned}$$

(3.22)

which is a contradiction because

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{z'_i}\right) >{\hat{L}}={1\over {\hat{t}}} \end{aligned}$$

(3.23)

with ${1\over {\hat{t}}}>2-{\hat{t}}$ for ${\hat{t}}\in ]0,1[$.

Thus, we can conclude that the case (3.18) cannot happen.

In a similar way we argue in order to obtain a contradiction in the case (3.19). In fact, assume that (3.19) holds. Notice that there exists a sequence $(\zeta ''_i)_{i\in \mathbb {N}}$ such that $\zeta ''_i\in Z_{k_i}$ and $|z''_i-\zeta ''_i|=1$ $\forall i\in \mathbb {N}$.

As before, the minimality property (3.13) implies that

$$\begin{aligned} \lim _{i\rightarrow \infty } \frac{2\min \{\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z''_i}\right) ,\, \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta ''_{i}}\right) \}}{ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z''_i}\right) +\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta ''_{i}}\right) }= {\hat{t}} \end{aligned}$$

(3.24)

and

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\left[ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z''_i}\right) +\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta ''_{i}}\right) \right] = 2. \end{aligned}$$

(3.25)

As a consequence, we infer that

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\min \left\{ \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ z''_i}\right) ,\, \mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{ \zeta ''_{i}}\right) \right\} = {\hat{t}}, \end{aligned}$$

(3.26)

which is in contradiction with (3.19) because

$$\begin{aligned} \lim _{i\rightarrow \infty } k_i\mathop {\mathrm{meas}}\nolimits T^L_{k_i}\left( {1\over k_i}\, C_{z''_i}\right) <{1\over {\hat{L}}}={\hat{t}}. \end{aligned}$$

(3.27)

Thus, we can conclude that $\lim \limits _{i\rightarrow \infty }\mathcal{L}(T^L_{k_i})={\hat{L}}$ so the proof is complete. $\square$

Remark 3.3

The results we present in this paper concern the existence of solutions with a large number of nodal regions. In particular, when $\Omega \subset \mathbb {R}^n$ with $n=1$, these solutions must have, as a consequence, a large number of zeroes. In next propositions we show that the term w can be chosen in such a way that the sign of the solutions is related to the nodal regions of the eigenfunctions of the Laplace operator $-\Delta$ in $H^1_0(\Omega )$. In particular, if $n=1$ we show that for suitable terms w in $L^2(\Omega )$, problem (1.1) does not have solutions with a small number of zeroes: more precisely, we show that for all positive integer h there exists $w_h\in L^2(\Omega )$ such that every solution of problem (1.1) has at least h zeroes (it follows from Corollary 3.6).

Lemma 3.4

Let D be a piecewise smooth, bounded open subset of $\mathbb {R}^n$, $n\ge 1$, and $\lambda _1$ be the first eigenvalue of the Laplace operator $-\Delta$ in $H^1_0(D)$.

Let $g:D\times \mathbb {R}\rightarrow \mathbb {R}$ be a Carathéodory function such that

$$\begin{aligned} \inf \{g(x,t)-\lambda _1 t\ :\ x\in D, \ t\ge 0\}>0. \end{aligned}$$

(3.28)

Let $u\in H^1(D)$ be a weak solution of the equation

$$\begin{aligned} -\Delta u=g(x,u)\quad \text{ in } D. \end{aligned}$$

(3.29)

Then $\inf _D u<0$. Moreover, if

$$\begin{aligned} \sup \{g(x,t)-\lambda _1 t\ :\ x\in D, \ t\le 0\}<0, \end{aligned}$$

(3.30)

then $\sup _D u>0$.

Proof

Let $e_1$ be a positive eigenfunction corresponding to the eigenvalue $\lambda _1$, that is

$$\begin{aligned} \Delta e_1+\lambda _1 e_1=0,\quad e_1>0\ \text{ in } D,\quad e_1\in H^1_0(D). \end{aligned}$$

(3.31)

Arguing by contradiction, assume that (3.28) holds and $u\ge 0$ in D. Then, from (3.29) we infer that

$$\begin{aligned} -\int _D\Delta u\, e_1\, dx=\int _D g(x,u) e_1\, dx \end{aligned}$$

(3.32)

which implies

$$\begin{aligned} \int _D Du\, De_1\, dx=\int _{\partial D} u\, De_1\cdot \nu \, d\sigma -\int _D u\, \Delta e_1\, dx =\int _D g(x,u) e_1\, dx, \end{aligned}$$

(3.33)

where $\nu$ denotes the outward normal on $\partial D$, so that

$$\begin{aligned} \int _{\partial D}u\, De_1\cdot \nu \, d\sigma \le 0, \end{aligned}$$

(3.34)

and $g(x,t)\ge \lambda _1t+c$ $\forall x\in D$, $\forall t\in \mathbb {R}$ for a suitable constant $c>0$. It follows that

$$\begin{aligned} \lambda _1\int _D u e_1\, dx\ge \int _D g(x,u) e_1dx\ge \lambda _1\int _Du e_1\, dx+c\int _De_1\, dx, \end{aligned}$$

(3.35)

which implies $c\int _D e_1\, dx\le 0$, that is a contradiction. Thus, the function u cannot be a.e. nonnegative in D.

In a similar way one can show that we cannot have $u\le 0$ a.e. in D when (3.30) holds, so the proof is complete. $\square$

In particular, Lemma 3.4 may be used to obtain informations on the effect of the term w on the sign changes of the solutions of problem (1.1), as we describe in the following proposition.

Proposition 3.5

Let $\Omega \subset \mathbb {R}^n$ with $n\ge 1$ and $e_k\in H^1_0(\Omega )$ be an eigenfunction of the Laplace operator $-\Delta$ with eigenvalue $\lambda _k$, that is $\Delta e_k+\lambda _k e_k=0$ in $\Omega$. Assume that $w\in L^2(\Omega )$ satisfies

$$\begin{aligned} e_kw\ge 0\ \text{ in } \Omega ,\quad \inf _\Omega |w|>\max \{\lambda _k t-t^p\ :\ t\ge 0\}. \end{aligned}$$

(3.36)

Let $\Omega _k\subseteq \Omega$ be a nodal region of $e_k$, that is $e_{k|_{\Omega _k}}\in H^1_0(\Omega _k)$, $e_k\ne 0$ everywhere in $\Omega _k$ and the sign of $e_k$ is constant in $\Omega _k$.

Then, there exists no function u in $H^1(\Omega )$ satisfying

$$\begin{aligned} u\, e_k\ge 0\ \text{ and } -\Delta u=|u|^{p-1}u+w\quad \text{ in } \Omega _k. \end{aligned}$$

(3.37)

In particular, if w satisfies (3.36), every solution u of problem (1.1) must satisfy

$$\begin{aligned} \inf _{\Omega _k}u<0\ \text{ if } \ e_k>0\ \text{ in } \ \Omega _k\ \text{ and } \ \sup _{\Omega _k}u>0\ \text{ if } \ e_k<0\ \text{ in } \ \Omega _k. \end{aligned}$$

(3.38)

Proof

Notice that $\lambda _k$ is the first eigenvalue of the Laplace operator $-\Delta$ in $H^1_0(\Omega _k)$ and $|e_k|$ is a corresponding positive eigenfunction. Moreover, if we set $g(x,t)=|t|^{p-1}t+w(x)$, we infer from (3.36) that, if $w(x)>0$,

$$\begin{aligned} g(x,t)\ge \lambda _kt+{\tilde{c}}\qquad \forall t\ge 0 \end{aligned}$$

(3.39)

and, if $w(x)<0$,

$$\begin{aligned} g(x,t)\le \lambda _kt-{\tilde{c}}\qquad \forall t\le 0 \end{aligned}$$

(3.40)

where

$$\begin{aligned} {\tilde{c}}=\inf _\Omega |w|-\max \{\lambda _k t-t^p\ :\ t\ge 0\}>0. \end{aligned}$$

(3.41)

Since $u e_k\ge 0$ and $w e_k\ge 0$ in $\Omega _k$ and $e_k$ has constant sign in $\Omega _k$, we have $u\ge 0$ and $w>0$ in $\Omega _k$ if $e_k>0$ in $\Omega _k$ and $u\le 0$, $w<0$ in $\Omega _k$ in the opposite case. Thus, our assertion follows from Lemma 3.4. In fact, for example, in the case $e_k>0$ in $\Omega _k$ we cannot have

$$\begin{aligned} -\Delta u=|u|^{p-1}u+w\quad \text{ in } \Omega _k \end{aligned}$$

(3.42)

otherwise $\inf _{\Omega _k} u<0$, because of Lemma 3.4, while $u\ge 0$ in $\Omega _k$.

In the opposite case, when $e_k<0$ in $\Omega _k$, one can argue in a similar way, so the proof is complete. $\square$

Corollary 3.6

Assume $n=1$ and $\Omega =]a,b[$. Let us denote by $\lambda _1<\lambda _2<\lambda _3<\ldots$ the eigenvalues of the Laplace operator $-\Delta$ in $H^1_0(]a,b[)$ and, for all $k\in \mathbb {N}$, consider an eigenfunction $e_k$ with eigenvalue $\lambda _k$. Moreover, assume that $w\in L^2(]a,b[)$ satisfy condition (3.36).

Set $h=k-1$ (h is the number of zeroes of $e_k$ in ]a, b[).

Then, every solution of problem (1.1) has in ]a, b[ at least h zeroes $\zeta _1,\zeta _2,\ldots ,\zeta _h$ such that

$$\begin{aligned} \left| a+\frac{b-a}{k}\, i-\zeta _i\right| <\frac{b-a}{k}\qquad \text{ for } i=1,\ldots ,h. \end{aligned}$$

(3.43)

Proof

Notice that the points $\nu _i=a+\frac{b-a}{k}\, i$, for $i=0,1,\ldots ,k$, are the zeroes of $e_k$ in [a, b] and the intervals $I_i=]\nu _{i-1},\nu _i[$, for $i=1,\ldots ,k$, are the nodal regions of $e_k$.

Assume, for example, that $e_k>0$ on $I_1$ (in a similar way one can argue if $e_k<0$ in $I_1$). Then, from Proposition 3.5 we infer that for every solution u of problem (1.1) we have $\inf _{I_i}u<0$ for i odd and $\sup _{I_i} u >0$ for i even.

Therefore, the function u has at least h zeroes $\zeta _1,\ldots ,\zeta _h$ such that $\zeta _i\in ]\nu _{i-1},\nu _{i+1}[$ for $i=1,\ldots ,h$, so the proof is complete. $\square$

Remark 3.7

Notice that all the assertions in Proposition 3.5 and Corollary 3.6 still hold when the nonlinear term $|u|^{p-1}u$ is replaced by $c_+(u^+)^p-c_-(u^-)^p$ where $c_+$ and $c_-$ are two positive constants. In this case we have only to replace $\max \{\lambda _kt-t^p\ :\ t\ge 0\}$ by $\max \{\lambda _k t-{\bar{c}} t^p\ :\ t\ge 0\}$, where ${\bar{c}}=\min \{c_+,c_-\}>0$.

Notice that this method to construct solutions with nodal regions having a check structure may be used for nonlinear elliptic problems with different boundary conditions, for systems and also when the nonlinear term has critical growth. For example, for all $\lambda \in \mathbb {R}$ consider the Dirichlet problem

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

(3.44)

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

$$\begin{aligned} \mathcal{F}(u)={1\over 2}\int _\Omega |{\nabla }u|^2dx-{n-2\over 2n}\int _\Omega |u|^{2n\over n-2}dx-{\lambda \over 2} \int _\Omega u^2dx-\int _\Omega w\, u\, dx\qquad u\in H^1_0(\Omega ).\nonumber \\ \end{aligned}$$

(3.45)

Using this method, if the functional $\mathcal{F}$ satisfies condition (2.53), one can prove that for $n\ge 4$ and $\lambda >0$ the functional $\mathcal{F}$ has an unbounded sequence of critical levels. More precisely, the following theorem can be proved.

Theorem 3.8

Let $n\ge 4$, $\lambda >0$, $w\in L^2(\Omega )$ and assume that condition (2.53) holds for the functional $\mathcal{F}$. Then, there exists ${\bar{k}}\ge k_\Omega$ such that for all $k\ge \bar{k}$ there exists $T^{{\bar{L}}}_k\in \mathcal{C}_{{\bar{L}}}(P_k,{\overline{\Omega }})$ and a solution $u_k$ of problem (3.44) such that $T^{\bar{L}}_k(P_k)={\overline{\Omega }}$ and, if for all $z\in Z_k$ we set $u^z_k(x)=u_k(x)$ when $x\in T^{{\bar{L}}}_k\left( {1\over k}\, C_z\right)$, $u^z_k(x)=0$ otherwise, then we have $[\sigma (z)u^z_k]^+\not \equiv 0$,

$$\begin{aligned} \mathcal{F}(u^z_k)\le {1\over n}\, S^{n/2}\quad \forall z\in Z_k \quad \text{ and } \quad \lim _{k\rightarrow \infty } \min \left\{ \mathcal{F}(u^z_k)\ :\ z\in Z_k\right\} = {1\over n}\, S^{n/2}, \end{aligned}$$

(3.46)

where S (the best Sobolev constant) is defined by

$$\begin{aligned} S=\inf \left\{ \int _{\mathbb {R}^n}|{\nabla }u|^2dx\ :\ u\in H^1(\mathbb {R}^n),\ \int _{\mathbb {R}^n}|u|^{2n\over n-2}dx=1\right\} . \end{aligned}$$

(3.47)

Let us point out that Theorem 3.8 gives a new result also when $w\equiv 0$ in $\Omega$. In fact, in this case the functional $\mathcal{F}$ is even but well known results (see [12, 16, 38]) guarantee only the existence of a finite number of solutions (because some compactness conditions hold only at suitable levels of $\mathcal{F}$). On the contrary our method, combined with some estimates as in [12] and in [16], allows us to construct infinitely many solutions with many nodal regions and arbitrarily large energy level.