Multiple solutions for a problem with discontinuous nonlinearity

Abstract

In this work, we use the Lusternik–Schnirelmann category to estimate the number of nontrivial solutions for a problem with discontinuous nonlinearity and subcritical growth.

1 Introduction

The present work studies the existence of multiple solutions for the following class of discontinuous problems

where $\Omega \subset \mathbb {R}^N$($N \ge 3$) is a smooth bounded domain, $f_{p, \delta }: \mathbb {R}\rightarrow \mathbb {R}$ is the odd function given by

$$\begin{aligned} f_{p, \delta }(t) = {\left\{ \begin{array}{ll} t|t|^{p-2}, &{} t \in [0, a],\\ (1 + \delta )t|t|^{p-2}, &{} t > a. \end{array}\right. } \end{aligned}$$

with $a, \delta > 0$ and $p \in (2, 2^*)$.

In [14], Benci and Cerami have considered the existence of multiple positive solutions for the case $\delta =0$, that is, for the problem

By using variational methods combined with the Lusternik–Schnirelmann category, Benci and Cerami proved that if p is close to $2^{*}=\frac{2N}{N-2}$, then problem $(P_{p,0})$ has at least $cat(\Omega )$ of positive solutions. Here, we recall that if X is a topological space and $A \subset X$ a closed subspace, we denote by ${\text {cat}}_X(A)$ the Lusternik–Schnirelmann category of A in X. The Lusternik–Schnirelmann category, $cat_{X}(A)$, is the least number of closed and contractible sets in X which cover A. If $X=A$, we use the short notation cat(X). Later, Benci and Cerami [15] generalized their previous result by working with a more general nonlinearity and Morse theory.

The reader can find in the literature a lot of papers where the existence and multiplicity of solutions for related problems to $(P_{p,0})$ are directly associated with the topological richness of $\Omega$, see Alves and Ding [9], Bahri and Coron [13], Rey [29], Struwe [31] and their references.

For the case $\delta >0$, the function $f_{p,\delta }$ is discontinuous and the study of existence of solution for $(P_{p,\delta })$ is totally different of the case $\delta =0$, because we cannot use directly the results for $C^{1}$-functionals, then the existence and multiplicity of solution for $(P_{p,\delta })$ associated with the topological richness of $\Omega$ is an open and interesting problem. Motivated by this fact, in the present paper we prove a result of multiplicity of solutions in the same spirit of [14]; more precisely, we prove that if $\delta$ is small enough and p is close to $2^*$, the problem $(P_{p,\delta })$ has at least $cat(\Omega )$ of positive solutions, see Theorem 1.1.

The interest in the study of nonlinear partial differential equations with discontinuous nonlinearities has increased because many free boundary problems arising in mathematical physics may be stated in this form. Among these problems, we have the obstacle problem, the seepage surface problem, and the Elenbaas equation; see, for example, [18,19,20].

A rich literature is available by now on problems with discontinuous nonlinearities, and we refer the reader to Ambrosetti and Turner [2], Ambrosetti et al. [5], Alves et al. [6], Alves and Bertone [7], Alves et al. [8], Badiale and Tarantelo [12], Carl et al. [16], Clarke [17], Chang [18], Carl and Dietrich [21], Carl and Heikkila [22, 23], Cerami [24], Hu et al. [25], Montreanu and Vargas [27], Radulescu [28] and their references. Several techniques have been developed or applied in their study, such as variational methods for nondifferentiable functionals, lower and upper solutions, global branching, fixed point theorem, and the theory of multivalued mappings.

Our main result is the following:

Theorem 1.1

There are $\delta ^* > 0$ and $p^* \in (2, 2^*)$ such that for each $\delta \in (0, \delta ^*)$ and $p \in (p^*, 2^*)$, $(P_{p,\delta })$ has at least ${\text {cat}}(\Omega )$ nontrivial solutions.

In the proof of the above result, we will adapt for our case an approach explored by Ambrosetti and Badiale [4]. The main idea consists in setting a suitable single function and then considering a dual functional, which is $C^{1}$ and their critical points produce solutions for $(P_{p,\delta })$. For more details, see Sect. 2.

Notations In this paper, we use the following notations:

• For $q \in (2, 2^*)$, we define $q'$ as the conjugate exponent of q, that is, $q' = \frac{q}{q-1}.$

• We denote by $2^+$ the conjugate exponent of $2^{*}=\frac{2N}{N-2}$, that is, $2^{+}=\frac{2N}{N+2}$.

• The usual norm of the Lebesgue spaces $L^t(\Omega )$ for $t \in [1,\infty ]$ will be denoted by $|.|_{t}$ and the norm of the Sobolev space $H^1_0(\Omega )$, by $\Vert .\Vert$;

• C denotes (possibly different) any positive constant.

• If $A \subset \mathbb {R}^N$ is a measurable set, we denote by meas(A) its Lebesgue measure.

• If X and Y are topological spaces, we say that X and Y are homotopically equivalent if there exist continuous functions $h:X \rightarrow Y$ and $q:Y \rightarrow X$ such that $q \circ h=id_X$ and $h \circ q=id_Y$.

2 An auxiliary problem

In the sequel, we consider the energy functional $I_{p, \delta }:H^1_0(\Omega ) \rightarrow \mathbb {R}$ associated with $(P_{p,\delta })$ given by

$$\begin{aligned} I_{p, \delta }(u) = \frac{1}{2}\int _{\Omega } |\nabla u|^2 \mathrm{d}x- \int _{\Omega }F_{p, \delta }(u)\mathrm{d}x, \end{aligned}$$

where

$$\begin{aligned} F_{p, \delta }(t) = \int _0^t f_{p, \delta }(r)\mathrm{d}r. \end{aligned}$$

Notice that $I_{p, \delta }$ is not a differentiable functional, because $F_{p, \delta }$ is only a continuous function. This fact does not allow to use the traditional methods to get multiplicity of solutions by using Lusternik–Schnirelmann category. To avoid this difficulty, we will adapt for our problem an approach explored in Ambrosetti and Badiale [4].

In what follows, we denote by $g_{p',\delta }: \mathbb {R}\rightarrow \mathbb {R}$ the odd function given by

$$\begin{aligned} g_{p', \delta }(s) = {\left\{ \begin{array}{ll} s|s|^{p' - 2}, &{} s \in [0, a^{p-1}],\\ a, &{} s \in [a^{p-1}, (1+\delta )a^{p-1}],\\ (1+\delta )^{-\frac{1}{p-1}}s |s|^{p' - 2}, &{} s \in [(1+\delta )a^{p-1},+\infty ). \end{array}\right. } \end{aligned}$$

The functions $f_{p, \delta }$ and $g_{p', \delta }$ are related in the following way:

(a):

$$\begin{aligned} f_{p, \delta }(g_{p', \delta }(s)) = {\left\{ \begin{array}{ll} s,&{} s \notin [a^{p-1}, (1+\delta )a^{p-1}],\\ a^{p-1}, &{} s \in [a^{p-1}, (1+\delta )a^{p-1}]; \end{array}\right. } \end{aligned}$$

(b):

$g_{p', \delta }(f_{p, \delta }(t)) = t, \forall t \in \mathbb {R}$.

In the sequel, $G_{p', \delta }$ denotes the primitive of $g_{p', \delta }$, that is,

$$\begin{aligned} G_{p', \delta }(s) := \int _0^s g_{p', \delta }(r)\mathrm{d}r. \end{aligned}$$

From definition of $g_{p', \delta }$, $G_{p', \delta }$ is an even function with

$$\begin{aligned} G_{p', \delta }(s) = {\left\{ \begin{array}{ll} \displaystyle \frac{1}{p'}s^{p'}, &{} s \in [0, a^{p-1}],\\ as - \displaystyle \frac{a^{p}}{p}, &{} s \in [a^{p-1},(1 + \delta )a^{p-1}],\\ \displaystyle \frac{\gamma _{\delta }}{p'} s^{p'} + \delta \displaystyle \frac{a^{p}}{p}, &{} s \in [(1+\delta )a^{p-1},+\infty ), \end{array}\right. } \end{aligned}$$

(2.1)

for $\gamma _\delta = (1+\delta )^{-\frac{1}{p-1}}$. Thus,

$$\begin{aligned} \gamma _\delta |s|^{\frac{1}{p-1}} \le |g_{p', \delta }(s)| \le |s|^{\frac{1}{p-1}}, \quad \forall s \in \mathbb {R}, \end{aligned}$$

(2.2)

and

$$\begin{aligned} \frac{\gamma _\delta }{p'}|s|^{p'} \le G_{p', \delta }(s) \le \frac{1}{p'} |s|^{p'}, \quad \forall s \in \mathbb {R}. \end{aligned}$$

(2.3)

To simplify the notation, we denote by $g_{p'}$ and $G_{p'}$ the functions $g_{p',0}$ and $G_{p',0}$, respectively.

The next step is to define the dual functional associated with $I_{p, \delta }$. By [26, Theorem 11.3], we know that for each $u \in L^{p'}(\Omega )$ there is an unique solution $w \in W_0^{1, p'}(\Omega ) \cap W^{2, p'}(\Omega )$ for the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w = u, \quad x \in \Omega ,\\ w = 0, \quad x \in \partial \Omega . \end{array}\right. } \end{aligned}$$

(2.4)

Moreover, there is a positive constant C independent of w such that

$$\begin{aligned} \Vert w\Vert _{W^{2,p'}(\Omega )} \le C|u|_{p'}. \end{aligned}$$

The above information permits to define a linear operator $K_{p', \Omega }: L^{p'}(\Omega ) \rightarrow W^{2,p'}(\Omega )$, such that for $u \in L^{p'}(\Omega )$, $K_{p', \Omega }(u)$ is the unique solution of (2.4). Hence,

$$\begin{aligned} \Vert K_{p', \Omega }(u)\Vert _{W^{2,p'}(\Omega )} \le C |u|_{p'}, \quad \forall u \in L^{p'}(\Omega ), \end{aligned}$$

from where it follows that $K_{p', \Omega }$ is continuous. On the other hand, since the embeddings below

$$\begin{aligned} W^{2,p'}(\Omega ) \hookrightarrow L^{s}(\Omega ), \quad \forall s \in \left[ 1,(p')^*\right) , \end{aligned}$$

are compact for

$$\begin{aligned} (p')^*= \left\{ \begin{array}{ll} \frac{Np'}{N-2p'}, &{} N > 2p', \\ +\infty , &{} 1\le N \le 2p' , \end{array} \right. \end{aligned}$$

we can ensure that $K_{p', \Omega }:L^{p'}(\Omega ) \rightarrow L^{p}(\Omega )$ is a linear compact operator, because $p \in (2,2^*)$ if, and only $p \in \left( 2,(p')^*\right)$. Moreover, it is easy to check that

$$\begin{aligned} \int _{\Omega }K_{p', \Omega }(u)v \mathrm{d}x = \int _{\Omega }K_{p', \Omega }(v)u \mathrm{d}x, \quad \forall u, v \in L^{p'}(\Omega ). \end{aligned}$$

(2.5)

Using the above notations, we set the functional $J_{p', \delta }: L^{p'}(\Omega ) \rightarrow \mathbb {R}$ given by

$$\begin{aligned} J_{p', \delta }(u) = \int _{\Omega }G_{p', \delta }(u) \mathrm{d}x - \frac{1}{2}\int _{\Omega }K_{p', \Omega }(u)u \mathrm{d}x. \end{aligned}$$

The functional $J_{p', \delta }$ is called the Dual functional associated with $I_{p, \delta }$. Observe that, differently of $I_{p, \delta }$, $J_{p', \delta } \in C^1(L^{p'}(\Omega ), \mathbb {R})$ and

$$\begin{aligned} J'_{p', \delta }(u)v&= \int _{\Omega } \Big (g_{p', \delta }(u) \mathrm{d}x - K_{p', \Omega }(u)\Big )v\ \mathrm{d}x, \forall u,v \in L^{p'}(\Omega ). \end{aligned}$$

Thus, $u \in L^{p'}(\Omega )$ is a critical point of $J_{p', \delta }$ if, and only if,

$$\begin{aligned} g_{p', \delta }(u) = K_{p', \Omega }(u) \text{ a.e. } \text{ in } \Omega . \end{aligned}$$

The above equality permits to prove the following proposition:

Proposition 2.1

If u is a critical point of $J_{p', \delta }$, then $v := g_{p', \delta }(u)$ is a solution of the problem $(P_{p,\delta })$.

Proof

If u is a critical point for $J_{p', \delta }$, then

$$\begin{aligned} v(x) = K_{p', \Omega }(u(x)) \text{ a.e. } \text{ in } \Omega , \end{aligned}$$

from where it follows that

$$\begin{aligned} -\Delta v(x) = u(x) \quad \text{ a.e. } \text{ in } \quad \Omega . \end{aligned}$$

Thereby, if $|v(x)| \ne a$,

$$\begin{aligned} -\Delta v(x) = u(x) = f_{p, \delta }(g_{p', \delta }(u(x))) = f_{p, \delta }(v(x)). \end{aligned}$$

If $|v(x)| = a$, we have that

$$\begin{aligned} -\Delta v(x)=0, \quad \text{ a.e. } \text{ in } \quad \mathcal {A}=\{x \in \Omega \,:\, |v(x)|=a \}. \end{aligned}$$

(2.6)

On the other hand, $v(x)=g_{p',\delta }(u(x))$ and if $|v(x)|=a$ then necessarily $u(x) \not =0$, by definition of $g_{p',\delta }$. Then,

$$\begin{aligned} -\Delta v(x)=u(x)\not =0 \quad \text{ a.e. } \text{ in } \quad \mathcal {A}. \end{aligned}$$

(2.7)

From (2.6)–(2.7), it follows that $\mathcal {A}$ has measure zero. Therefore,

$$\begin{aligned} -\Delta v(x) = f_{p,\delta }(v(x)), \quad \text{ a.e } \text{ in } \quad \Omega \quad \text{ and } \quad v \in H^{1}_{0}(\Omega ). \end{aligned}$$

Now, the elliptic regularity gives $v \in W^{2,\frac{p}{p-1}}(\Omega )$, showing that v is a solution of $(P_{p,\delta })$.   $\square$

Motivated by the last proposition, we will look for critical points of $J_{p', \delta }$. The result below establishes that $J_{p', \delta }$ satisfies the mountain pass geometry.

Proposition 2.2

The functional $J_{p', \delta }$ has the mountain pass geometry, that is,

(i):

$J_{p', \delta }(0) = 0$ and there is $\rho > 0$ such that

$$\begin{aligned} \inf _{|u|_{p'} = \rho } J_{p', \delta }(u) > 0 \quad \text{ and } \quad J_{p', \delta }(u) \ge 0, \ \forall u \in L^{p'}(\Omega ) \quad \text{ with } \quad |u|_{p'} \le \rho . \end{aligned}$$

(ii):

There is $\psi \in L^{p'}(\Omega )$ such that

$$\begin{aligned} |\psi |_{p'} > \rho \quad \text{ and } \quad J_{p', \delta }(\psi ) < 0. \end{aligned}$$

Proof

We begin by showing (i). The equality $J_{p', \delta }(0) = 0$ is immediate. From (2.3),

$$\begin{aligned} \int _{\Omega } G_{p', \delta }(u)\mathrm{d}x \ge \frac{\gamma _\delta }{p'} |u|^{p'}_{p'}, \ \forall u \in L^{p'}(\Omega ), \end{aligned}$$

(2.8)

and by Hölder’s inequality and continuity of $K_{p', \Omega }$, there is $C>0$ such that

$$\begin{aligned} \int _{\Omega }K_{p', \Omega }(u)u \ \mathrm{d}x \le C |u|^2_{p'}, \quad \forall u \in L^{p'}(\Omega ). \end{aligned}$$

(2.9)

Thus, (2.8) and (2.9) combine to give

$$\begin{aligned} J_{p', \delta }(u)\ge & {} \frac{\gamma _\delta }{p'} |u|^{p'}_{p'} - \frac{C}{2} |u|^2_{p'}\\= & {} |u|^{p'}_{p'} \left( \frac{\gamma _\delta }{p'} - \frac{C}{2}|u|^{2 - p'}_{p'} \right) . \end{aligned}$$

Since $p' < 2$, there is $\rho > 0$ as in (i). For (ii), notice that for each $\tilde{\psi }\in C_0(\Omega ) {\setminus } \{0\}$,

$$\begin{aligned} \lim _{t \rightarrow \infty } J_{p', \delta }(t\tilde{\psi }) = - \infty . \end{aligned}$$

Therefore, for $t_0 > 0$ large enough, $\psi := t_0 \tilde{\psi }$ is as required in (ii). $\square$

The next proposition is crucial in our argument, because it proves that $J_{p', \delta }$ verifies the (PS) condition for $\delta$ small enough.

Proposition 2.3

There is $\delta _0 > 0$ such that for all $\delta \in [0, \delta _0]$, the functional $J_{p', \delta }$ satisfies the (PS) condition, that is, if $(u_n) \subset L^{p'}(\Omega )$ is such that

$$\begin{aligned} \sup _{n \in \mathbb {N}} |J_{p', \delta }(u_n)| < \infty \ \text{ and } \ J'_{p', \delta }(u_n) \rightarrow 0 \text{ as } n \rightarrow \infty , \end{aligned}$$

then there is $u \in L^{p'}(\Omega )$ such that, up to a subsequence, $u_n \rightarrow u$ in $L^{p'}(\Omega )$.

Proof

Let $(u_n) \subset L^{p'}(\Omega )$ be a sequence with

$$\begin{aligned} \sup _{n \in \mathbb N} |J_{p', \delta }(u_n)| < \infty ~~ \text{ and } ~~ J_{p', \delta } '(u_n) \rightarrow 0. \end{aligned}$$

Taking a subsequence if necessary, we can assume that $J_{p', \delta }(u_n) \rightarrow d$ as $n \rightarrow \infty$, and so $(u_n)$ is a bounded sequence in $L^{p'}(\Omega )$. Indeed, for n large enough,

$$\begin{aligned} d + 1 + |u_n|_{p'} \ge J_{p', \delta }(u_n) - \frac{1}{2} J'_{p', \delta }(u_n)u_n = \int _{\Omega } (G_{p', \delta }(u_n) - \frac{1}{2}g_{p', \delta }(u_n)u_n)\mathrm{d}x.\quad \end{aligned}$$

(2.10)

As $g_{p', \delta }$ and $G_{p', \delta }$ are odd and even functions, respectively, (2.2) and (2.3) ensure that

$$\begin{aligned} G_{p',\delta }(t) - \frac{1}{2}t g_{p', \delta }(t) \ge \left( \frac{\gamma _\delta }{p'} - \frac{1}{2} \right) |t|^{p'}, \quad \forall t \in \mathbb {R}. \end{aligned}$$

Once $p' < 2$ and $\gamma _\delta = (1+\delta )^{-\frac{1}{p-1}}$, there is $\delta _0 > 0$ such that

$$\begin{aligned} \left( \frac{\gamma _\delta }{p'} - \frac{1}{2} \right) >0, \quad \forall \delta \in [0, \delta _0]. \end{aligned}$$

Thereby, by (2.10),

$$\begin{aligned} d + 1 + |u_n|_{p'} \ge \left( \frac{\gamma _\delta }{p'} - \frac{1}{2} \right) |u_n|^{p'}_{p'}, \end{aligned}$$

from where it follows that $(u_n)$ is a bounded sequence. As $L^{p'}(\Omega )$ is a reflexive space, there is $u \in L^{p'}(\Omega )$ such that $u_n \rightharpoonup u$ weakly in $L^{p'}(\Omega )$. Then, by compactness of $K_{p', \Omega }$,

$$\begin{aligned} K_{p', \Omega }(u_n) \rightarrow K_{p', \Omega }(u) \text{ in } L^p(\Omega ) \text{ as } n \rightarrow \infty . \end{aligned}$$

(2.11)

On the other hand, the limit $J'_{p', \delta }(u_n) \rightarrow 0$ in $(L^{p'}(\Omega ))'=L^p(\Omega )$ yields

$$\begin{aligned} g_{p', \delta }(u_n) - K_{p', \Omega }(u_n) \rightarrow 0 \quad \text{ in } \quad L^p(\Omega ). \end{aligned}$$

So, by (2.11),

$$\begin{aligned} g_{p', \delta }(u_n) \rightarrow K_{p', \Omega }(u) =: w \text{ in } L^p(\Omega ) \text{ as } n \rightarrow \infty . \end{aligned}$$

(2.12)

Then there is $h \in L^p(\Omega )$ such that

$$\begin{aligned}&|g_{p', \delta }(u_n(x))| \le h(x), \quad \forall n \in \mathbb {N}, \end{aligned}$$

(2.13)

$$\begin{aligned}&g_{p', \delta }(u_n(x)) \rightarrow w(x) \text{ a.e. } \text{ in } \Omega . \end{aligned}$$

(2.14)

Let

$$\begin{aligned} \Gamma := \{ x \in \Omega ; |w(x)| = a \} \text{ and } \tilde{\Omega }:= \Omega \backslash \Gamma . \end{aligned}$$

We claim that $u_n \rightarrow f_{p, \delta }(w)$ in $L^{p'}(\tilde{\Omega })$. If $x \in \tilde{\Omega }$, we have

$$\begin{aligned} (f_{p,\delta }\circ g_{p',\delta })(u_n(x)) \rightarrow f_{p,\delta }(w(x)), \end{aligned}$$

and also $|u_n(x)| \not \in [a^{p-1},(1+\delta )a^{p-1}]$ for n large enough, hence $(f_{p,\delta }\circ g_{p',\delta })(u_n(x))=u_n(x)$, so $u_n(x) \rightarrow f_{p,\delta }(w(x))$. Combining (2.13) and the fact that $f_{p, \delta }$ is odd and increasing, one easily derives a uniform estimate in $L^{p'}(\tilde{\Omega })$ for sequence $(u_n)$, so by Dominated Convergence Theorem the conclusion follows.

On the other hand, by using the same type of arguments found [4, Theorem 1], it is possible to show that $meas(\Gamma )=0$. Then, the above analysis leads to

$$\begin{aligned} u_n \rightarrow u \quad \text{ in } \quad L^{p'}(\Omega ), \end{aligned}$$

and the proposition is proved. $\square$

We finish this section by proving that $J_{p', \delta }$ has a nontrivial critical point

Theorem 2.4

The mountain pass level of $J_{p', \delta }$, denoted by $c_{p',\delta }$, is a critical level.

Proof

Propositions 2.2 and 2.3 permit to apply the Mountain Pass Theorem found in [1]. Then, there is a critical $u_{p', \delta } \in L^{p'}(\Omega )$ whose the energy is equal to mountain pass level of $J_{p', \delta }$, that is,

$$\begin{aligned} J'_{p', \delta }(u_{p', \delta }) = 0 \quad \text{ and } \quad J_{p', \delta }(u_{p', \delta })= c_{p', \delta }. \end{aligned}$$

$\square$

3 Nehari manifold associated with $\varvec{J_{p', \delta }}$

In this section, we will make a careful study of the Nehari manifold $\mathcal N_{p', \delta }$ associated with $J_{p', \delta }$ given by

$$\begin{aligned} \mathcal N_{p', \delta }&:= \{ u \in L^{p'}(\Omega )\backslash \{0\} ; J'_{p', \delta }(u)u = 0 \} \\&= \{ u \in L^{p'}(\Omega )\backslash \{0\} ; \int _\Omega g_{p', \delta }(u)u \mathrm{d}x= \int _\Omega K_{p', \Omega }(u)u\mathrm{d}x\}. \end{aligned}$$

It is worth pointing out that since $g_{p', \delta }$ is not a $C^1$ function, we cannot assert that $\mathcal N_{p', \delta }$ is a differentiable manifold. This fact brings for us some difficulties to apply Lagrange Multiplier on $\mathcal N_{p', \delta }$. However, we overcome this difficulty by adapting for our problem some arguments found in Szulkin and Weth [30].

Our first lemma follows by using the continuity of $K_{p', \Omega }$ together with the inequality $G_{p',\delta }(t)-\frac{1}{2}tg_{p',\delta }(t) \ge C|t|^{p'}$ for some constant $C>0$, and it has the following statement

Lemma 3.1

There is $\eta = \eta (p) > 0$ such that

$$\begin{aligned} |u|_{p'}, J_{p', \delta }(u) > \eta , \quad \forall u \in \mathcal N_{p', \delta }. \end{aligned}$$

The second result can be obtained by using the same arguments found [32, Chapter 4], because the function $g_{p',\delta }$ is odd, $\frac{g_{p',\delta }(t)}{t}$ is decreasing for $t>0$ and $K_{p', \Omega }$ is a linear operator.

Lemma 3.2

For each $v \in L^{p'}(\Omega ) \backslash \{0\}$, there is an unique $t_v > 0$ such that

$$\begin{aligned} J'_{p', \delta }(t_v v)(t_v v) = 0. \end{aligned}$$

(3.1)

Moreover,

$$\begin{aligned} c_{p',\delta }=\inf _{u \in \mathcal N_{p', \delta }}J_{p',\delta }(u). \end{aligned}$$

As an immediate consequence of the last lemma is the following corollary

Corollary 3.3

If u is a critical point of $J_{p',\delta }$ with $u^{\pm } \not = 0$, then $J_{p',\delta }(u) \ge 2 c_{p',\delta }$.

Proof

The proof follows with the same type of arguments found in [10, Section 4] or [11, Theorem 2.4]. $\square$

The next lemma is crucial in our approach, because it guarantees the continuity of the function $v \mapsto t_v$ in $L^{p'}(\Omega )\backslash \{0\}$.

Lemma 3.4

For $(u_n)\subset L^{p'}(\Omega )$ and $u \in L^{p'}(\Omega )\backslash \{0\}$, let $t_{u_n}, t_u > 0$ be as in (3.1). If $u_n \rightarrow u$ in $L^{p'}(\Omega )$, then $t_{u_n} \rightarrow t_u$.

Proof

For simplicity, set $t_n := t_{u_n}$. First of all, note that $t_n \not \rightarrow 0$. Indeed, taking $v = u_n$ in (3.1) and using (2.2) and (2.9), we get

$$\begin{aligned} \gamma _\delta |u_n|^{p'}_{p'}t_n^{p'} \le \int _\Omega g_{p', \delta }(t_n u_n)t_n u_n \mathrm{d}x= \int _\Omega K_{p', \Omega } (t_nu_n)(t_nu_n) \mathrm{d}x\le Ct_n^2|u_n|^2_{p'}, \end{aligned}$$

for some $C > 0$. So, for some $c > 0$,

$$\begin{aligned} c |u_n|^{p'-2}_{p'} \le t_n^{2-p'}, \end{aligned}$$

and the desired property follows from the fact that $p' \in (1,2)$ and $(u_n)$ are a bounded sequence in $L^{p'}(\Omega )$.

Moreover, $(t_n)$ is bounded. In fact, the continuity of $K_{p', \Omega }$, (2.2) and (3.1) leads to

$$\begin{aligned} t_n^{p'-2} C|u_n|^{p'}_{p'} \ge \frac{1}{t^2_n} \int _\Omega g_{p', \delta }(t_n u_n)t_n u_n \mathrm{d}x= \int _\Omega K_{p', \Omega } (u_n)(u_n) \mathrm{d}x\rightarrow \int _\Omega K_{p', \Omega }(u)u\mathrm{d}x>0, \end{aligned}$$

which implies the boundedness of $(t_n)$.

Finally, up to a subsequence, we have $t_n \rightarrow t_0$. Then, by Lebesgue’s Theorem,

$$\begin{aligned} \int _\Omega g_{p', \delta }(t_0 u) t_0 u \mathrm{d}x =&\lim _{n} \int _{\Omega } g_{p', \delta }(t_n u_n)t_n u_n\mathrm{d}x\\ =&\lim _{n} \int _\Omega K_{p', \Omega }(t_n u_n) t_n u_n \mathrm{d}x= \int _\Omega K_{p', \Omega } (t_0u)t_0u\mathrm{d}x. \end{aligned}$$

Now, the uniqueness of $t_u$ ensures that $t_u = t_0 = \displaystyle \lim _{n \rightarrow +\infty }t_n$. $\square$

In the sequel, without loss of generality we assume that $0 \in \Omega$ and denote by $w_{p, r} \in H^1_0(B_r(0))$ be a positive ground-state solution of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w = |w|^{p-2}w, \quad x\in B_r(0), \\ w = 0, \quad x \in \partial B_r(0), \end{array}\right. } \end{aligned}$$

where $r > 0$ is such that the sets

$$\begin{aligned} \Omega ^+ := \{x \in \mathbb {R}^N ; {\text {dist}}(x, \Omega ) \le r\}, \quad \Omega ^- := \{x \in \Omega ; {\text {dist}}(x, \partial \Omega ) \ge r\} \end{aligned}$$

and $\Omega$ are homotopically equivalent. Hence,

$$\begin{aligned} I_{p, B_r(0)}(w_{p,r})=b_{p,B_r(0)} \quad \text{ and } \quad I'_{p, B_r(0)}(w_{p,r})=0, \end{aligned}$$

where

$$\begin{aligned} I_{p, B_r(0)}(u) = \frac{1}{2}\int _{B_r(0)} |\nabla u|^2 \mathrm{d}x- \frac{1}{p}\int _{B_r(0)}|u|^{p}\mathrm{d}x, \quad \forall u \in H^{1}_0(B_r(0)) \end{aligned}$$

and $b_{p,B_r(0)}$ denotes the mountain pass level associated with $I_{p, B_r(0)}$. It is well known that $w_{p, r}$ is radially symmetric and of class $C^2$. Therefore, $u_{p', r} := w_{p, r}^{p-1}$ is positive, radially symmetric and a critical point of the functional $J_{p',B_r(0)}: L^{p'}(B_r(0)) \rightarrow \mathbb {R}$ given by

$$\begin{aligned} J_{p', B_r(0)}(u)&= \int _{B_r(0)}G_{p'}(u)\mathrm{d}x - \frac{1}{2} \int _{B_r(0)}K_{p', B_r(0)}(u)u\mathrm{d}x\\&= \frac{1}{p'}\int _{B_r(0)} |u|^{p'}\mathrm{d}x - \frac{1}{2} \int _{B_r(0)}K_{p', B_r(0)}(u)u\mathrm{d}x. \end{aligned}$$

Moreover,

$$\begin{aligned} J_{p',B_r(0)}(u_{p', r})&= \frac{1}{p'}\int _{B_r(0)} |u_{p', r}|^{p'} \mathrm{d}x - \frac{1}{2}\int _{B_r(0)} K_{p', B_r(0)}(u_{p', r})u_{p', r} \mathrm{d}x \nonumber \\&= \frac{1}{p'}\int _{B_r(0)} |u_{p', r}|^{p'} \mathrm{d}x- \frac{1}{2}\int _{B_r(0)} |u_{p', r}|^{p'} \mathrm{d}x \nonumber \\&= \left( \frac{1}{p'} - \frac{1}{2} \right) \int _{B_r(0)}|u_{p', r}|^{p'} \mathrm{d}x = \left( \frac{1}{2} - \frac{1}{p} \right) \int _{B_r(0)}|u_{p', r}|^{p'} \mathrm{d}x \nonumber \\&= \left( \frac{1}{2} - \frac{1}{p} \right) \int _{B_r(0)}|w_{p, r}|^{p} \mathrm{d}x =I_{p,B_r(0)}(w_{p, r})=b_{p,B_r(0)}. \end{aligned}$$

(3.2)

Arguing as in [3], it is possible to prove that $b_{p,B_r(0)}=c_{p',B_r(0)}$, where $c_{p',B_r(0)}$ denotes the mountain pass level of $J_{p',B_r(0)}$.

In the sequel, let $\Phi _{p', \delta }: \Omega ^- \rightarrow \mathcal N_{p', \delta }$ be the map defined by

$$\begin{aligned} \Phi _{p', \delta }(y)(x) = {\left\{ \begin{array}{ll} t_{p', y}u_{p', r}(|x - y|), &{} x \in B_r(y),\\ 0, &{} x \in \Omega \backslash B_r(y), \end{array}\right. } \end{aligned}$$

where $t_{p', y} > 0$ is such that $t_{p', y}u_{p', r}(|. - y|) \in \mathcal N_{p', \delta }$, for each $y \in \Omega ^-$. Using the function $\Phi _{p', \delta }(y)$, we are able to prove that

$$\begin{aligned} c_{p', \delta } \le c_{p',B_r(0)}. \end{aligned}$$

(3.3)

Indeed, firstly it is very important to observe that from definition of $\Omega ^-$ and r, we have that $B_r(y) \subset \Omega$ for all $y \in \Omega ^-$, consequently $supp(\Phi _{p', \delta }(y))=B_r(y) \subset \Omega$. Then,

$$\begin{aligned} c_{p', \delta }&\le J_{p', \delta }(\Phi _{p', \delta }(y)) = J_{p', \delta }(t_{p', y}u_{p', r})\\&= \int _{\Omega }G_{p', \delta }(t_{p', y}u_{p', r})\mathrm{d}x - \frac{t_{p', y}^2}{2}\int _{\Omega } K_{p', \Omega }(u_{p', r})u_{p', r}\mathrm{d}x. \end{aligned}$$

By the maximum principle,

$$\begin{aligned} K_{p', \Omega }(u_{p', r}) > K_{p', B_r(0)}(u_{p', r}) \ \text{ on } \ B_r(0), \end{aligned}$$

and so,

$$\begin{aligned} c_{p', \delta }&\le J_{p', \delta }(\Phi _{p', \delta }(y)) \le \int _{\Omega }G_{p', \delta }(t_{p', y}u_{p', r})\mathrm{d}x - \frac{1}{2}\int _{\Omega } K_{p', \Omega }(t_{p', y}u_{p', r})t_{p', y} u_{p', r}\mathrm{d}x \nonumber \\&\le \int _{B_r(0)}G_{p'}(t_{p', y}u_{p', r})\mathrm{d}x - \frac{1}{2}\int _{B_r(0)} K_{p', B_r(0)}(t_{p', y} u_{p', r})t_{p', y} u_{p', r} \mathrm{d}x\nonumber \\&= J_{p', B_r(0)}(t_{p', y}u_{p', r})\nonumber \le J_{p', B_r(0)}(u_{p', r}) =c_{p',B_r(0)}, \nonumber \end{aligned}$$

which proves (3.3).

Our next result shows the behavior of the levels $c_{p', \delta }$ and $c_{p',B_r(0)}$ with respect to the numbers p and $\delta$.

Proposition 3.5

The following limits hold:

$$\begin{aligned} \lim _{p\rightarrow 2^*, \delta \rightarrow 0} c_{p', \delta } = \lim _{p \rightarrow 2^*} c_{p',B_r(0)} = c_* := \frac{1}{N}S^{N/2}, \end{aligned}$$

where S is the best constant for the embedding $H_0^1(\Omega ) \hookrightarrow L^{2^*}(\Omega )$.

Proof

We begin by showing the second limit. Let us denote by $I_p$ and $J_{p'}$ the functionals $I_{p,0}$ and $J_{p',0}$, respectively. If $b_{p,,B_r(0)}$ denotes the mountain pass level of $I_{p,,B_r(0)}$, adapting the arguments found in [3], it follows $b_{p,B_r(0)}=c_{p',B_r(0)}$. Moreover, in [14] it was proved that

$$\begin{aligned} b_{p,B_r(0)} = \left( \frac{1}{2} - \frac{1}{p} \right) m_{p,r}^{\frac{p}{p-2}}, \end{aligned}$$

with

$$\begin{aligned} \displaystyle {m_{p,r} := \inf \limits _{w \in H^1_0(B_r(0))\backslash \{0\}} \frac{\int _{B_r(0)} |\nabla w|^2 \mathrm{d}x}{\left( \int _{B_r(0)} |w|^{p} \mathrm{d}x\right) ^{2/p}} } \quad \text{ and } \quad \lim \limits _{p \rightarrow 2^*}m_{p,r} = S. \end{aligned}$$

Therefore,

$$\begin{aligned} \lim _{p \rightarrow 2^*} c_{p',B_r(0)} = \lim _{p \rightarrow 2^*} b_{p,B_r(0)} = \lim _{p \rightarrow 2^*} \left( \frac{1}{2} - \frac{1}{p} \right) m_{p,r}^{\frac{p}{p-2}} = c_*. \end{aligned}$$

(3.4)

Here, we would like to point out that the above arguments could be made with $B_r(0)$ replaced by $\Omega$, because the result found in [14] still holds for $\Omega$. Then, if $c_{p'}=c_{p',0}$, we also have

$$\begin{aligned} \lim _{p \rightarrow 2^*} c_{p'} = c_*. \end{aligned}$$

(3.5)

Now, we deal with the first limit in the statement, that is,

$$\begin{aligned} \lim _{p\rightarrow 2^*, \delta \rightarrow 0} c_{p', \delta } =c_*. \end{aligned}$$

(3.6)

Let $(\delta _n), (p_n)$ be sequences satisfy $\delta _n \rightarrow 0$ and $p_n \rightarrow 2^*$ as $n \rightarrow \infty$. By Theorem 2.4, for each $n \in \mathbb {N}$ there is $u_n \in L^{p'_n}(\Omega )$ such that

$$\begin{aligned} J_{p_n', \delta _n}(u_n) = c_{p_n', \delta _n} \quad \text{ and } \quad J'_{p_n', \delta _n}(u_n) = 0. \end{aligned}$$

Setting $t_n > 0$ be such that $t_nu_n \in \mathcal N_{p'_n} := \mathcal N_{p'_n,0}$, we find

$$\begin{aligned} c_{p_n'} \le J_{p_n'}(t_n u_n) = J_{p_n', \delta _n}(t_n u_n) + \int _{\Omega } \left[ G_{p_n'}(t_nu_n) - G_{p_n', \delta _n}(t_nu_n)\right] \mathrm{d}x. \end{aligned}$$

(3.7)

Claim 3.6

$$\begin{aligned} \int _{\Omega }\left[ G_{p_n'}(t_nu_n) - G_{p_n', \delta _n}(t_nu_n)\right] ~\mathrm{d}x = o_n(1). \end{aligned}$$

(3.8)

Indeed, by the definitions of $G_{p', \delta }$ and $G_{p'}$,

$$\begin{aligned} 0 \le G_{p'}(s) - G_{p', \delta }(s) \le (1 - \gamma _\delta )\frac{1}{p'}|s|^{p'}, \quad \forall s \in \mathbb {R}, \end{aligned}$$

and so,

$$\begin{aligned} 0 \le \int _{\Omega } \left[ G_{p_n'}(t_nu_n) - G_{p_n', \delta _n} (t_nu_n)\right] \mathrm{d}x \le (1 - {\gamma _{\delta _n}}) \frac{1}{p'_n}t_n^{p_n'}\int _{\Omega }|u_n|^{p'_n} \ \mathrm{d}x. \end{aligned}$$

(3.9)

From this, we will get the desired conclusion by showing that $(|t_nu_n|_{p_n'})$ is bounded, since $\gamma _{\delta _n} \rightarrow 1$ as $n \rightarrow \infty$. Firstly, from inequality

$$\begin{aligned} G_{p',0}(t) \ge G_{p',\delta }(t), \quad \forall t \in \mathbb {R}, \end{aligned}$$

we have that

$$\begin{aligned} J_{p',0}(u) \ge J_{p',\delta }(u) \quad \forall u \in L^{p'}(\Omega ), \end{aligned}$$

implying that $c_{p_n'} \ge c_{p_n', \delta _n}$ for all $n \in \mathbb {N}$. From this, $(|u_n|_{p_n'})$ is bounded, because $(c_{p_n'})$ is a bounded sequence and

$$\begin{aligned} c_{p_n'}&\ge c_{p_n', \delta _n} = J_{p_n', \delta _n}(u_n) = J_{p_n', \delta _n}(u_n) - \frac{1}{2}J'_{p_n', \delta _n}(u_n)(u_n)\\&= \int _{\Omega } \left[ G_{p_n', \delta _n}(u_n) - \frac{1}{2} g_{p_n', \delta _n}(u_n)u_n\right] \ \mathrm{d}x \ge \left( \frac{\gamma _{\delta _n}}{p_n'} - \frac{1}{2}\right) \int _{\Omega } |u_n|^{p'}\mathrm{d}x. \end{aligned}$$

Next, we will work to show that $(t_n)$ is also a bounded sequence. To this end, we need to prove that

$$\begin{aligned} \liminf _{n \rightarrow \infty } |u_n|_{p_n'} > 0. \end{aligned}$$

(3.10)

As $p'_n > 2^+$ and $|\Omega | < \infty$, it follows that

$$\begin{aligned} K_{p_n', \Omega }(u_n) = K_{2^+, \Omega }(u_n), \ \forall n \in \mathbb {N}. \end{aligned}$$

As $u_n \in \mathcal N_{p'_n, \delta _n}$, the above equality combined with Hölder inequality gives

$$\begin{aligned} \gamma _{\delta }|u_n|^{p'_n}_{p'_n} \le \int _{\Omega } g_{p_n', \delta _n}(u_n)u_n~\mathrm{d}x = \int _{\Omega } K_{2^+, \Omega } (u_n)u_n~\mathrm{d}x \le C |u_n|_{2^+}^2 \le C |\Omega |^{\frac{2}{\theta _n}} |u_n|_{p_n'}^2, \end{aligned}$$

where $\displaystyle {\frac{1}{2^+} = \frac{1}{p_n'} + \frac{1}{\theta _n}}$. Then,

$$\begin{aligned} 1 \le C |\Omega |^{\frac{2}{\theta _n}}|u_n|_{p_n'}^{2-p_n'}. \end{aligned}$$

Once $\theta _n \rightarrow \infty$ and $p_n' \rightarrow 2^+$ as $p_n \rightarrow 2^*$, the last inequality implies that there is $\kappa > 0$ such that $|u_n|_{p_n'} > \kappa$ for n large enough, which proves (3.10).

Now, by using the fact $u_n \in \mathcal N_{p_n', \delta _n}$ together with (2.2) and (3.10), we obtain

$$\begin{aligned} \int _{\Omega } K_{p'_n, \Omega }(u_n)u_n \mathrm{d}x = \int _{\Omega } g_{p_n', \delta _n}(u_n)u_n \mathrm{d}x \ge \gamma _{\delta _n}\int _{\Omega } |u_n|^{p_n'} \ \mathrm{d}x > \kappa , \end{aligned}$$

(3.11)

for n large enough. Hence,

$$\begin{aligned} \frac{c_*}{2}\le c_{p_n'} \le J_{p_n'}(t_n u_n) = \frac{t_n^{p'_n}}{p'_n} \int _{\Omega } |u_n|^{p'_n} \ \mathrm{d}x - \frac{t^2_n}{2} \int _{\Omega } K_{p'_n, \Omega }(u_n)u_n \mathrm{d}x, \end{aligned}$$

for n large enough. Gathering the boundedness of $(|u_n|_{p'_n})$ with (3.11), we derive

$$\begin{aligned} \frac{c_*}{2}\le c t_n^{p'_n} - \tau t^2_n, \ \forall n \in \mathbb {N}, \end{aligned}$$

from where it follows that $(t_n)$ is bounded. From this, $(|t_n u_n|_{p_n'})$ is bounded, which finishes the proof of Claim 3.6. Therefore from (3.3) and (3.7)

$$\begin{aligned} c_{p_n'} \le c_{p_n', \delta _n} + o_n(1) \le c_{p_n'} + o_n(1). \end{aligned}$$

As $(p_n)$ and $(\delta _n)$ are arbitrary sequences, (3.5) gives

$$\begin{aligned} \lim _{p \rightarrow 2^*, \delta \rightarrow 0} c_{p', \delta } = c_*. \end{aligned}$$

$\square$

The next step would be to determine whether or not the restriction of $J_{p', \delta }$ to $\mathcal N_{p', \delta }$ satisfies the (PS)-condition. The standard approach would lead us to the study of the second derivative of $J_{p', \delta }$, which we cannot compute, since this functional is not twice differentiable. With this in mind, we will adapt for our case some ideas explored in [30].

Consider the application

$$\begin{aligned} \hat{m}_{p', \delta }:L^{p'}(\Omega ) {\setminus } \{0\} \rightarrow \mathcal N_{p', \delta } \end{aligned}$$

given by

$$\begin{aligned} \hat{m}_{p', \delta } (u) = t_u u, \end{aligned}$$

where $t_u$ is defined by (3.1). Using the above notations, it is possible to prove that

• (a) $\hat{m}_{p', \delta }$ is a continuous application.

• (b) There is $\tau > 0$ such that $t_u > \tau$, $\forall u \in \mathcal {S}_{p'}=\{u \in L^{p'}(\Omega )\,:\,|u|_{p'}=1\}$; Indeed, if $(u_n) \subset L^{p'}(\Omega )$ is such that $t_{u_n} \rightarrow 0$ as $n \rightarrow \infty$, then $t_{u_n} u_n \rightarrow 0$ as $n \rightarrow \infty$. This contradicts Lemma 3.1, since $t_{u_n}u_n \in \mathcal N_{p', \delta }$.

• (c) Given $\mathcal W \subset \mathcal S_{p'}$ compact, there is $C_\mathcal W > 0$ such that $C_\mathcal W > t_u$, $\forall u \in \mathcal W$; In fact, this is a consequence of the continuity of the application $v \mapsto t_v$, as shown in Lemma 3.4.

In the sequel, we consider the application $m_{p', \delta }: \mathcal S_{p'} \rightarrow \mathcal N_{p', \delta }$, the restriction of $\hat{m}_{p', \delta }$ to the sphere $\mathcal S_{p'}$. Observe that $m_{p', \delta }$ is a homeomorphism, with inverse given by

$$\begin{aligned} m_{p', \delta }^{-1}(u) = \frac{u}{|u|_{p'}}, \quad \forall u \in \mathcal N_{p', \delta }. \end{aligned}$$

Let us also consider the application $\hat{\Psi }_{p', \delta }: L^{p'}(\Omega ) {\setminus } \{0\} \rightarrow \mathbb {R}$ given by

$$\begin{aligned} \hat{\Psi }_{p', \delta }(u) := J_{p', \delta }(\hat{m}_{p', \delta }(u)), \end{aligned}$$

and its restriction to the sphere, $\Psi _{p', \delta }: \mathcal S_{p'} \rightarrow \mathbb {R}$. Note that both $\hat{\Psi }_{p', \delta }$ and $\Psi _{p', \delta }$ are continuous. The following result is crucial in our approach and its proof can be found in [30, Chapter 3].

Lemma 3.7

The applications defined above satisfy:

(i):

$\hat{\Psi }_{p', \delta } \in C^1(L^{p'}(\Omega ) {\setminus } \{0\}, \mathbb {R})$ and, for $u \in L^{p'}(\Omega )\backslash \{0\}$,

$$\begin{aligned} \hat{\Psi }'_{p', \delta }(u)v= & {} \frac{|\hat{m}_{p', \delta }(u)|_{p'}}{|u|_{p'}} J_{p', \delta }'(\hat{m}_{p', \delta }(u))v,\\= & {} t_u J_{p', \delta }'(\hat{m}_{p', \delta }(u))v, \quad \forall v \in L^{p'}(\Omega ); \end{aligned}$$

(ii):

$\Psi _{p', \delta } \in C^1(\mathcal S_{p'}, \mathbb {R})$ and, for $u \in \mathcal S_{p'}$,

$$\begin{aligned} \Psi '_{p', \delta }(u)v = |m_{p', \delta }(u)|_{p'} J_{p', \delta }'(m_{p', \delta }(u))v, \quad \forall v \in T_u\mathcal S_{p'}, \end{aligned}$$

where $T_u\mathcal S_{p'}$ denotes the tangent space of $\mathcal S_{p'}$ at u;

(iii):

If $(u_n) \subset \mathcal S_{p'}$ is a (PS) sequence for $\Psi _{p', \delta }$, then $(m_{p', \delta }(u_n))$ is a (PS) sequence for $J_{p', \delta }$; if $(u_n) \subset \mathcal N_{p', \delta }$ is a (bounded) (PS) sequence for $J_{p',\delta }$, then $(m^{-1}_{p', \delta }(u_n))$ is a (PS) sequence for $\Psi _{p', \delta }$;

(iv):

$u \in \mathcal S_{p'}$ is a critical point of $\Psi _{p', \delta }$ if, and only if, $m_{p', \delta }(u)$ is a (nonzero) critical point of $J_{p', \delta }$. Moreover,

$$\begin{aligned} \inf \limits _{\mathcal S_{p'}} \Psi _{p', \delta } = \inf \limits _{\mathcal N_{p', \delta }} J_{p', \delta }. \end{aligned}$$

Corollary 3.8

$\Psi _{p', \delta }$ is bounded from below and satisfies the (PS) condition.

Proof

The boundedness follows by a combination of the previous result and Lemma 3.1. On the (PS) condition, let $(u_n) \subset \mathcal S_{p'}$ be a (PS) sequence for $\Psi _{p', \delta }$. Thus, by Lemma 3.7-(iii), $\big (m_{p', \delta }(u_n)\big )$ is a (PS) sequence for $J_{p', \delta }$. By Proposition 2.3, $\big (m_{p', \delta }(u_n)\big )$ has a strongly convergent subsequence. Since $m_{p', \delta }$ is a homeomorphism, $(u_n)$ has a strongly convergent subsequence, that is, $\Psi _{p', \delta }$ satisfies the (PS) condition. $\square$

Before concluding this section, we will show that Palais–Smale sequence of $J_{2^+}$ produces a Palais–Smale sequence for $I_{2^*}:H^{1}_0(\Omega ) \rightarrow \mathbb {R}$ given by

$$\begin{aligned} I_{2^*}(u)=\frac{1}{2}\int _{\Omega }|\nabla u|^{2}\,\mathrm{d}x -\frac{1}{2^*}\int _{\Omega }|u|^{2^*}\,\mathrm{d}x. \end{aligned}$$

Lemma 3.9

If $(u_n) \subset \mathcal N_{2^+}$ is a $(PS)_d$ sequence of $J_{2^+}$, then there is $t_n >0$ such that $v_n=t_n|u_n|^{2^+-2}u_n \in \mathcal M_{2^*}$, where $\mathcal M_{2^*}$ is the Nehari manifold associated with $I_{2^*}$. Moreover, the sequence $(v_n)$ is a $(PS)_d$ sequence of $I_{2^*}$.

Proof

Let $(u_n) \subset \mathcal N_{2^+}$ be a $(PS)_d$ sequence for $J_{2^+}$, that is, $(u_n)$ satisfies

$$\begin{aligned} J'_{2^+}(u_n)u_n = 0, \quad J_{2^+}(u_n) = d + o_n(1) \quad \text{ and } \quad \Vert J'_{2^+}(u_n)\Vert _{L^{2^*}(\Omega )} = o_n(1). \end{aligned}$$

(3.12)

Then $(u_n) \subset L^{2^+}(\Omega )$ is a bounded sequence. In what follows, we define $(w_n) \subset H^1_0(\Omega )$ and $(v_n) \subset L^{2^*}(\Omega )$ by $w_n := K_{2^+, \Omega }(u_n)$ and $\tilde{v}_n := |u_n|^{2^+-2}u_n$, for each $n \in \mathbb {N}$. So $u_n = |\tilde{v}_n|^{2^*-2}\tilde{v}_n$ and $w_n$ is the unique solution of the problem

$$\begin{aligned} \left\{ \begin{array}{l} -\Delta w_n = u_n, \quad x \in \Omega ,\\ w_n = 0, \quad \text{ on } \quad \partial \Omega . \end{array} \right. \end{aligned}$$

(3.13)

By (3.12) and the definitions of $w_n$ and $v_n$, both $(w_n), (\tilde{v}_n)$ are bounded sequences with $|\tilde{v}_n - w_n|_{2^*} \rightarrow 0$ as $n \rightarrow \infty$, that is, $w_n = \tilde{v}_n + o_n(1)$ in $L^{2^*}(\Omega )$.

The sequence $(w_n)$ is a $(PS)_d$ sequence for $I_{2^*}$. Indeed, for any $\phi \in H^1_0(\Omega )$, (3.13) gives

$$\begin{aligned} I'_{2^*}(w_n)\phi&= \int _{\Omega } \nabla w_n \nabla \phi \ \mathrm{d}x - \int _{\Omega } |w_n|^{2^*-2}w_n\phi \ \mathrm{d}x\\&= \int _{\Omega } u_n \phi \ \mathrm{d}x - \int _{\Omega } |w_n|^{2^*-2}w_n\phi \ \mathrm{d}x\\&= \int _{\Omega } \left( |\tilde{v}_n|^{2^*-2}\tilde{v}_n- |w_n|^{2^*-2}w_n\right) \phi \ \mathrm{d}x\\&\le C\big ||\tilde{v}_n|^{2^*-2}\tilde{v}_n - |w_n|^{2^*-2}w_n\big |_{2^+} \Vert \phi \Vert \end{aligned}$$

and so,

$$\begin{aligned} \Vert I'_{2^*}(u_n)\Vert _{H^{-1}(\Omega )} \le C\big ||\tilde{v}_n|^{2^*-2}\tilde{v}_n - |w_n|^{2^*-2}w_n\big |_{2^+} \end{aligned}$$

As $w_n-\tilde{v}_n \rightarrow 0$ in $L^{2^{*}}(\Omega )$, it follows that

$$\begin{aligned} \Vert I'_{2^*}(w_n)\Vert _{H^{-1}(\Omega )} = o_n(1). \end{aligned}$$

(3.14)

Moreover, since $J'_{2^+}(u_n)u_n = 0$,

$$\begin{aligned} I_{2^*}(w_n)&= \frac{1}{2}\int _{\Omega }|\nabla w_n|^2 \ \mathrm{d}x - \frac{1}{2^*}\int _{\Omega }|w_n|^{2^*}\mathrm{d}x\nonumber \\&= \frac{1}{2}\int _{\Omega }w_nu_n\ \mathrm{d}x - \frac{1}{2^*} \int _{\Omega }|v_n|^{2^*} \mathrm{d}x + o_n(1)\nonumber \\&= \frac{1}{2}\int _{\Omega }w_nu_n\ \mathrm{d}x - \frac{1}{2^*} \int _{\Omega }|u_n|^{2^+} \mathrm{d}x + o_n(1)\nonumber \\&= \left( \frac{1}{2} - \frac{1}{2^*}\right) \int _{\Omega }|u_n|^{2^+} \mathrm{d}x + o_n(1)\nonumber \\&= \left( \frac{1}{2^+} - \frac{1}{2}\right) \int _{\Omega }|u_n|^{2^+} \mathrm{d}x + o_n(1)\nonumber \\&= J_{2^+}(u_n) + o_n(1) = d + o_n(1). \end{aligned}$$

(3.15)

Thus, $(w_n)$ is a $(PS)_d$ sequence for $I_{2^*}$. Next, fix $\tilde{w}_n := t_n w_n$ where $t_n > 0$ is such that $t_n w_n \in \mathcal M_{2^*}$. We claim that $(\tilde{w}_n)$ is a $(PS)_d$ for $I_{2^*}$. Indeed, once $(u_n)$ is a bounded sequence on $\mathcal N_{2^+}$, then $\liminf \limits _{n \rightarrow \infty }|u_n|_{2^+} > 0$. Furthermore, using again that $w_n=v_n +o_n(1)$ in $L^{2^*}(\Omega )$, we see that $t_n$ satisfies

$$\begin{aligned} t_n^2\int _{\Omega } |\nabla w_n|^2 \mathrm{d}x&= t_n^{2^*}\int _{\Omega }|w_n|^{2^*}\mathrm{d}x\\&= t_n^{2^*}\Big [\int _{\Omega }|v_n|^{2^*}\mathrm{d}x + o_n(1)\Big ]\\&= t_n^{2^*}\int _{\Omega }|u_n|^{2^+}\mathrm{d}x + t^{2^*}_n o_n(1), \end{aligned}$$

which leads to

$$\begin{aligned} (t^{2-2^*}_n - 1)\int _{\Omega } |u_n|^{2^+}\mathrm{d}x = o_n(1). \end{aligned}$$

Therefore, as $(u_n)$ does not vanish, $t_n \rightarrow 1$ as $n \rightarrow \infty$ which permits to conclude that $(\tilde{w}_n)$ is a $(PS)_d$ sequence for $I_{2^*}$. Hence, the sequence $(v_n)$ given by $v_n=t_n\tilde{v}_n=t_n|u_n|^{2^+-2}u_n$ is also a $(PS)_d$ sequence for $I_{2^*}$. $\square$

4 Proof of Theorem 1.1

After the study made in the previous section, we are able to prove our main result. To this end, we will consider the application $\beta : \mathcal N_{p', \delta } \rightarrow \mathbb {R}^N$ given by

$$\begin{aligned} \beta (u) = \frac{\displaystyle \int _{\Omega } x |u|^{2^+} \mathrm{d}x}{\displaystyle \int _{\Omega } |u|^{2^+}\mathrm{d}x}. \end{aligned}$$

Once $L^{p'}(\Omega ) \hookrightarrow L^{2^+}(\Omega )$, $\beta$ is well defined and

$$\begin{aligned} \beta \circ \Phi _{p', \delta }(x) = x, \quad \forall x \in \Omega ^-. \end{aligned}$$

(4.1)

The next result establishes an important estimate associated with $\beta$.

Proposition 4.1

There are $\epsilon , p^*, \delta _1 > 0$ such that for each $p \in (p^*, 2^*)$ and $\delta \in (0, \delta _1)$, if $u \in \mathcal N_{p', \delta }$ satisfies $J_{p', \delta }(u) \le c_* + \epsilon$, then $\beta (u) \in \Omega ^+$, where $c_*$ is defined in Proposition 3.5.

Suppose by contradiction that the result is false. Then, there are sequences $(\epsilon _n), (p_n), (\delta _n)$ with $\epsilon _n \rightarrow 0$, $\delta _n \rightarrow 0$, $p_n \rightarrow 2^*$ and $u_n \in \mathcal N_{p'_n, \delta _n}$ such that

$$\begin{aligned} J_{p_n', \delta _n}(u_n) \le c_* + \epsilon _n \quad \text{ and } \quad \beta (u_n) \notin \Omega ^+. \end{aligned}$$

(4.2)

To simplify the notation, we will use $J_n := J_{p'_n, \delta _n}$, $\mathcal N_n := \mathcal N_{p'_n, \delta _n}$, $G_n: = G_{p'_n, \delta _n}$, $g_n := g_{p'_n, \delta _n}$ and $K_n := K_{p'_n, \Omega }$.

We begin noticing that $(|u_n|_{p'_n})$ is bounded, since (2.2) and (2.3) lead to

$$\begin{aligned} c_* + \epsilon _n&\ge J_n(u_n) = J_n(u_n) - \frac{1}{2}J'_n(u_n)u_n\\&= \int _{\Omega } \left( G_n(u_n) - \frac{1}{2}g_n(u_n)u_n\right) \mathrm{d}x\\&\ge \left( \frac{\gamma _{\delta _n}}{p'_n} - \frac{1}{2} \right) |u_n|_{p'_n}^{p'_n}, \end{aligned}$$

and $\left( \frac{\gamma _{\delta _n}}{p'_n} - \frac{1}{2} \right) \rightarrow \frac{1}{N}$ as $n \rightarrow \infty$.

Claim 4.2

There is $t_n > 0$ such that $t_nu_n \in \mathcal N_{p'_n}$, that is, $J'_{p'_n}(t_nu_n)t_nu_n = 0$, and $t_n \rightarrow 1$ as $n \rightarrow \infty$.

The existence of such $(t_n)$ is a consequence of the definition of $J_{p'_n}$. Thus, for each $n \in \mathbb {N}$,

$$\begin{aligned} \int _{\Omega }|t_nu_n|^{p'_n}\ \mathrm{d}x = \int _{\Omega } K_n(t_nu_n)t_nu_n\ \mathrm{d}x, \end{aligned}$$

that is,

$$\begin{aligned} t_n^{p'_n - 2}\int _{\Omega }|u_n|^{p'_n} \ \mathrm{d}x = \int _{\Omega } K_n(u_n)u_n \ \mathrm{d}x. \end{aligned}$$

(4.3)

Since $u_n \in \mathcal N_n$, (2.2) gives

$$\begin{aligned} \int _{\Omega } K_n (u_n)u_n \mathrm{d}x = \int _{\Omega } g_n(u_n)u_n \mathrm{d}x = \int _{\Omega }|u_n|^{p'_n} \mathrm{d}x + o_n(1). \end{aligned}$$

(4.4)

By (4.3) and (4.4),

$$\begin{aligned} \left( {t_n^{p'_n - 2}} - 1\right) \int _{\Omega }|u_n|^{p'_n}\ \mathrm{d}x = o_n(1). \end{aligned}$$

(4.5)

Moreover, by (2.3),

$$\begin{aligned} c_{p'_n, \delta _n}&\le J_n(u_n) = \int _{\Omega } (G_n(u_n) - \frac{1}{2}K_n(u_n)u_n)\ \mathrm{d}x\\&= \int _{\Omega }(G_n(u_n) - \frac{1}{2}g_n(u_n)u_n) \ \mathrm{d}x\\&\le \frac{1}{p'_n} \int _{\Omega }|u_n|^{p'_n}\ \mathrm{d}x. \end{aligned}$$

Then, by Proposition 3.5, $\liminf \limits _{n \rightarrow \infty }|u_n|^{p'_n}_{p'_n} > 0$. Thereby, (4.5) ensures that $\lim \limits _{n \rightarrow \infty }t_n = 1$ and the claim is proved.

Let $\tilde{u}_n := t_n u_n$, for all $n \in \mathbb {N}$. Since $\tilde{u}_n \in \mathcal N_{p'_n}$, by using (4.2) and the same argument explored in the proof of Claim 3.6, we get

$$\begin{aligned} c_* + o_n(1)&= c_{p'_n} \le J_{p'_n}(\tilde{u}_n)=J_{p'_n}(t_n u_n) \\&\le J_n(u_n) + o_n(1) \le c_* + \epsilon _n + o_n(1), \end{aligned}$$

that is,

$$\begin{aligned} \lim _{n \rightarrow \infty } J_{p'_n}(\tilde{u}_n) = c_*. \end{aligned}$$

Now observe that by the definition of $J_{2^+}$, there is $r_n > 0$ such that $r_n \tilde{u}_n \in \mathcal N_{2^+}$.

Claim 4.3

$(|\tilde{u}_n|_{p'_n})$ and $(r_n)$ are bounded sequences and $\liminf \limits _{n \rightarrow \infty } |r_n\tilde{u}_n|_{p'_n}^{p'_n} > 0$.

In fact, once $r_n \tilde{u}_n \in \mathcal N_{2^+}$,

$$\begin{aligned} \int _{\Omega }|r_n\tilde{u}_n|^{2^+}\ \mathrm{d}x = \int _{\Omega } K_{2^+, \Omega }(r_n \tilde{u}_n)r_n \tilde{u}_n \ \mathrm{d}x, \end{aligned}$$

that is,

$$\begin{aligned} \frac{1}{r_n^{2 - 2^+}} \int _{\Omega }|\tilde{u}_n|^{2^+}\ \mathrm{d}x = \int _{\Omega } K_{2^+, \Omega }(\tilde{u}_n)\tilde{u}_n \ \mathrm{d}x \end{aligned}$$

from where it follows that

$$\begin{aligned} \frac{1}{r_n^{2 - 2^+}} \int _{\Omega }|\tilde{u}_n|^{2^+}\ \mathrm{d}x = \int _{\Omega } K_{p'_n, \Omega }(\tilde{u}_n)\tilde{u}_n \ \mathrm{d}x. \end{aligned}$$

Here, we had used the fact that $K_{p'_n, \Omega }(\tilde{u}_n)=K_{2^*, \Omega }(\tilde{u}_n)$, because $p'_n > 2^+$ and $meas(\Omega ) < \infty$.

Besides, since $\tilde{u}_n \in \mathcal N_{p'_n}$ and $J_{p'_n}(\tilde{u}_n) \rightarrow c_*$,

$$\begin{aligned} \left( \frac{1}{2} - \frac{1}{p'_n}\right) \int _{\Omega }|\tilde{u}_n|^{p'_n}\ \mathrm{d}x = \left( \frac{1}{p_n} - \frac{1}{2}\right) \int _{\Omega }|\tilde{u}_n|^{p'_n}\ \mathrm{d}x = J_{p'_n}(\tilde{u}_n) = c_* + o_n(1), \end{aligned}$$

so

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega } |\tilde{u}_n|^{p'_n} \ \mathrm{d}x = N c_* = S^{N/2}. \end{aligned}$$

(4.6)

Since $\tilde{u}_n=t_n u_n$ and $t_n \rightarrow 1$, it follows that $(|u_n|_{p'_n})$ is also bounded. Using Hölder inequality,

$$\begin{aligned} \int _{\Omega }|\tilde{u}_n|^{2^+}\ \mathrm{d}x \le |\Omega |^{\frac{2^+}{\theta _n}} \left( \int _{\Omega }|u_n|^{p'_n}\ \mathrm{d}x\right) ^{\frac{2^+}{p'_n}}, \end{aligned}$$

where $\frac{1}{\theta _n} = \frac{1}{2^+} - \frac{1}{p'_n} \rightarrow 0$ as $n \rightarrow \infty$. Therefore,

$$\begin{aligned} \int _{\Omega }|\tilde{u}_n|^{2^+} \mathrm{d}x \le o_n(1) + \left( \int _{\Omega }|\tilde{u}_n|^{p'_n} \mathrm{d}x \right) ^{\frac{2^+}{p'_n}}, \end{aligned}$$

(4.7)

and $(|\tilde{u}_n|_{2^+})$ is bounded. Moreover, arguing as in the proof of (3.10) we have that $\displaystyle \liminf _{n \rightarrow +\infty }|\tilde{u}_n|_{2^+}^{2^+}>0$. This together with (4.6), (4.7) and the fact that $\tilde{u}_n\in \mathcal N_{p'_n}$ gives that $(r_n)$ is bounded. We finish the proof of the claim by applying Lemma 3.1.

Now, using the equality

$$\begin{aligned} c_*=\inf _{u \in \mathcal M_{2^*}}I_{2^*}(u)=\inf _{u \in \mathcal N_{2^+}}J_{2^+}(u), \quad (\text{ see } \,\,\, [2]) \end{aligned}$$

we find

$$\begin{aligned} c_* \le \ J_{2^+}(r_n\tilde{u}_n). \end{aligned}$$

Thus, combining the Hölder’s inequality with (4.7) and Claim 4.3, we obtain

$$\begin{aligned} c_* \le&\ J_{2^+}(r_n\tilde{u}_n) = \frac{1}{2^+}\int _{\Omega }|r_n\tilde{u}_n|^{2^+} \mathrm{d}x - \frac{1}{2}\int _{\Omega } K_{2^+, \Omega }(r_n \tilde{u}_n)r_n \tilde{u}_n \ \mathrm{d}x\\ \le&\ \left( \frac{1}{p'_n} + o_n(1)\right) \left[ o_n(1) + \left( \int _{\Omega }|r_n\tilde{u}_n|^{p'_n} \mathrm{d}x\right) ^{\frac{2^+}{p'_n}}\right] \\&\ - \frac{1}{2}\int _{\Omega } K_{p'_n, \Omega }(r_n \tilde{u}_n) r_n \tilde{u}_n\ \mathrm{d}x\\ =&\ o_n(1) + \frac{1}{p'_n}\left( \int _{\Omega }|r_n\tilde{u}_n|^{p'_n} \mathrm{d}x\right) ^{\frac{2^+}{p'_n}} - \frac{1}{2}\int _{\Omega } K_{p'_n, \Omega }(r_n \tilde{u}_n) r_n \tilde{u}_n\ \mathrm{d}x\\ =&\ o_n(1) + \frac{1}{p'_n}\left( \int _{\Omega }|r_n\tilde{u}_n|^{p'_n} \mathrm{d}x\right) ^{1+ o_n(1)} - \frac{1}{2}\int _{\Omega } K_{p'_n, \Omega }(r_n \tilde{u}_n) r_n \tilde{u}_n\ \mathrm{d}x\\ =&\ o_n(1) + \frac{1}{p'_n}\int _{\Omega }|r_n\tilde{u}_n|^{p'_n} \mathrm{d}x - \frac{1}{2}\int _{\Omega } K_{p'_n, \Omega }(r_n \tilde{u}_n) r_n \tilde{u}_n\ \mathrm{d}x\\ =&\ o_n(1) + J_{p'_n}(r_n \tilde{u}_n)\\ \le&\ o_n(1) + J_{p'_n}(\tilde{u}_n) = o_n(1) + c_*. \end{aligned}$$

Consequently, $w_n := r_n \tilde{u}_n \in \mathcal N_{2^+}$ satisfies $J_{2^+}(w_n) \rightarrow c_*$. Without loss of generality, using the Ekeland’s Variational Principle, we can assume that $w_n$ also satisfies $J'_{2^+}(w_n) \rightarrow 0$ as $n \rightarrow \infty$, that is, $(w_n)$ is a (PS) sequence for $J_{2^+}$ at the level $c_*$.

By Lemma 3.9, there is a (PS) sequence $(v_n) \subset \mathcal M_{2^*}$ for $I_{2^*}$ at the level $c_*$. Observe that $(v_n)$ satisfies

$$\begin{aligned} \frac{\int _{\Omega }|\nabla v_n|^2\mathrm{d}x}{\left( \int _{\Omega }{|v_n|^{2^*}\mathrm{d}x}\right) ^{\frac{2}{2^*}}} = \left( \int _{\Omega }{|v_n|^{2^*}\mathrm{d}x}\right) ^{1 - \frac{2}{2^*}} = \left( \int _{\Omega }{|v_n|^{2^*}\mathrm{d}x}\right) ^{\frac{2}{N}} \end{aligned}$$

(4.8)

and

$$\begin{aligned} c_* + o_n(1) = I_{2^*}(v_n) = \left( \frac{1}{2} - \frac{1}{2^*}\right) \int _{\Omega }|v_n|^{2^*}\mathrm{d}x = \frac{1}{N}\int _{\Omega }|v_n|^{2^*}\mathrm{d}x, \end{aligned}$$

that is,

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega }|v_n|^{2^*}\mathrm{d}x = Nc_* = S^{N/2}. \end{aligned}$$

(4.9)

By (4.8) and (4.9),

$$\begin{aligned} \displaystyle {\lim _{n \rightarrow \infty } \frac{\int _{\Omega }|\nabla v_n|^2\mathrm{d}x}{\left( \int _{\Omega }{|v_n|^{2^*}\mathrm{d}x}\right) ^{\frac{2}{2^*}}} = S.} \end{aligned}$$

By setting the function $w_n=\frac{v_n}{|v_n|_{2^*}}$, we have that

$$\begin{aligned} |w_n|_{2^*}=1 \quad \text{ and } \quad \lim _{n \rightarrow +\infty }\int _{\Omega }|\nabla w_n|^{2}\,\mathrm{d}x=S. \end{aligned}$$

Arguing as in [32, Lemma 5.23], we can apply the Concentration–Compactness Lemma due to Lions [32, Lema 1.40] to find $u \in D^{1,2}(\mathbb {R}^N)$ and a subsequence of $(u_n)$, still denoted by itself, such that

$$\begin{aligned} w_n&\rightharpoonup&u \quad \text{ in } \quad D^{1,2}(\mathbb {R}^N),\\ |\nabla w_n|^{2}&\rightharpoonup&\mu \quad \text{ in } \quad \mathcal {M}(\mathbb {R}^N), \end{aligned}$$

and

$$\begin{aligned} |w_n|^{2^*} \rightarrow \nu \quad \text{ in } \quad \mathcal {M}(\mathbb {R}^N), \end{aligned}$$

where $\mu$ and $\nu$ are positive finite measure with $\nu$ concentrated at a single point $y \in \overline{\Omega }$. Therefore,

$$\begin{aligned} \int _{\Omega } x|w_n|^{2^*}\mathrm{d}x \rightarrow \int _{\Omega }x\,d \nu =y \in \overline{\Omega } \end{aligned}$$

or equivalently

$$\begin{aligned} \alpha (v_n):={\frac{\displaystyle \int _{\Omega } x|v_n|^{2^*}\mathrm{d}x}{\displaystyle \int _{\Omega }|v_n|^{2^*}\mathrm{d}x}} \rightarrow \int _{\Omega }x\,d \nu =y \in \overline{\Omega }, \end{aligned}$$

implying that

$$\begin{aligned} \alpha (v_n) \in \Omega ^+ \end{aligned}$$

for n large enough. Thereby,

$$\begin{aligned} \beta (u_n) = \alpha (v_n) \in \Omega ^+, \end{aligned}$$

for n large enough, which contradicts (4.2). This completes the proof. $\square$

As a by-product of the last proposition, we have that

Corollary 4.4

For $\epsilon , p^*, \delta _1 > 0$ given in Proposition 4.1, for each $p \in (p^*, 2^*)$, $\delta \in (0, \delta _1)$, if $u \in \mathcal S_{p'}$ satisfies $\Psi _{p', \delta }(u) \le c_* + \epsilon$, then $\beta \big (m_{p', \delta }(u)\big ) \in \Omega ^+$.

Proof

Indeed, for fixed $p \in (p^*, 2^*), \delta \in (0, \delta _1)$, if $u \in \mathcal S_{p'}$ is such that $\Psi _{p', \delta }(u) \le c_* + \epsilon$, then $m_{p', \delta }(u) \in \mathcal N_{p', \delta }$ with $J_{p', \delta }\big (m_{p', \delta }(u)\big ) \le c_* + \epsilon$. By Theorem 4.1, $\beta \big (m_{p', \delta }(u)\big ) \in \Omega ^+$. $\square$

The next result establishes a crucial relation between ${\text {cat}}\big (\mathcal S_{p'}^{c_* + \epsilon }\big )$ and ${\text {cat}}(\Omega )$, where

$$\begin{aligned} \mathcal S_{p'}^{c_* + \epsilon }=\{u \in S_{p'}\,:\, \Psi _{p', \delta }(u) \le c_* + \epsilon \}. \end{aligned}$$

Proposition 4.5

For $\epsilon , p^*, \delta _1 > 0$ given in Proposition 4.1, $p \in (p^*, 2^*)$ and $\delta \in (0, \delta _1)$, we have

$$\begin{aligned} {\text {cat}}\big (\mathcal S_{p'}^{c_* + \epsilon }\big ) \ge {\text {cat}}(\Omega ). \end{aligned}$$

Proof

By Proposition 3.5, we can fix $r>0$ such that $c_{p',B_r(0)}<c_*+\epsilon$. Let $k = {\text {cat}}(\mathcal S_{p'}^{c_* + \epsilon })$. Then, there are k closed contractible sets $A_j \subseteq \mathcal S_{p'}^{c_* + \epsilon }$, $j = 1, \ldots , k$ such that $\cup _{j = 1}^kA_j = \mathcal S_{p'}^{c_* + \epsilon }$. This means that there are k continuous applications $h_j: [0,1]\times A_j \rightarrow \mathcal S_{p'}^{c_* +\epsilon }$ such that

$$\begin{aligned} h_j(0, u) = u, \quad h_j(1, u) = h_j(1, v), \forall u,v \in A_j, \, j = 1, \ldots , k. \end{aligned}$$

(4.10)

Setting $B_j := \big (m^{-1}_{p', \delta } \circ \Phi _{p', \delta }\big )^{-1}(A_j), j = 1, \ldots , k$, we derive that $B_j$ are closed and $B_j \subset \Omega ^-$. Moreover, as

$$\begin{aligned} \Psi _{p', \delta }((m^{-1}_{p', \delta } \circ \Phi _{p', \delta })(y))=J_{p',\delta }(\Phi _{p', \delta }(y))=c_{p',B_r(0)}<c_*+\epsilon , \forall y \in \Omega ^-, \end{aligned}$$

or equivalently

$$\begin{aligned} \Psi _{p', \delta }((m^{-1}_{p', \delta } \circ \Phi _{p', \delta })(\Omega ^-)) \subset \mathcal S_{p'}^{c_* + \epsilon }, \end{aligned}$$

we also have that

$$\begin{aligned} \bigcup \limits _{j = 1}^k B_j = \Omega ^-. \end{aligned}$$

(4.11)

Consider the applications $l_j : [0,1]\times B_j \rightarrow \Omega ^+$ given by

$$\begin{aligned} l_j(t, x) := \beta \circ m_{p', \delta }\circ h_j(t, m^{-1}_{p', \delta }\circ \Phi _{p', \delta }(x)). \end{aligned}$$

Then $l_j$ is continuous and, for $x, y \in B_j \subset \Omega ^+$, using (4.10) and (4.1),

$$\begin{aligned} l_j(0,x)&= \beta \circ m_{p', \delta }\circ h_j(0, m^{-1}_{p', \delta }\circ \Phi _{p', \delta }(x))\\&= \beta \circ m_{p', \delta }\circ m^{-1}_{p', \delta }\circ \Phi _{p', \delta }(x)\\&= \beta \circ \Phi _{p', \delta }(x) = x, \end{aligned}$$

and

$$\begin{aligned} l_j(1, x)&= \beta \circ m_{p', \delta }\circ h_j(1, m^{-1}_{p', \delta }\circ \Phi _{p', \delta }(x))\\&= \beta \circ m_{p', \delta }\circ h_j(1, m^{-1}_{p', \delta }\circ \Phi _{p', \delta }(y))\\&= l_j(1, y). \end{aligned}$$

Thus, $B_j$ are contractible and by (4.11),

$$\begin{aligned} cat(\Omega )={\text {cat}}_{\Omega ^+}(\Omega ^-) \le k = {\text {cat}}(\mathcal S_{p'}^{c_* + \epsilon }), \end{aligned}$$

as desired. $\square$

Proof of Theorem 1.1

Let $p \in (p^*,2^*)$ and $0< \delta < \delta ^* :=\min \{\delta _0, \delta _1\}$. Then, by Lemma 3.7-(iv), $c_* + \epsilon > c_* = \inf _{\mathcal S_{p'}}\Psi _{p', \delta }$. This and Corollary 3.8 allow us to apply the Lusternik–Schnirelmann category to $\Psi _{p', \delta }$, which guarantee us that $\Psi _{p', \delta }$ has at least ${\text {cat}}(\mathcal S_{p'}^{c_* +\epsilon })$ nontrivial critical points on $\mathcal S_{p'}^{c_* + \epsilon }$. Applying Lemma 3.7-(iv) and Proposition 4.5, we conclude that $J_{p', \delta }$ has at least ${\text {cat}}(\Omega )$ nontrivial critical points. Thus, by Theorem 2.1, $(P_{p,\delta })$ has at least ${\text {cat}}(\Omega )$ nontrivial solutions. Moreover, since f is odd, Corollary 3.3 together with maximum principle yields these solutions can be chosen positive. $\square$