Asymptotically linear fractional p-Laplacian equations

Abstract

In this paper we study the multiplicity of weak solutions to (possibly resonant) nonlocal equations involving the fractional p-Laplacian operator. More precisely, we consider a Dirichlet problem driven by the fractional p-Laplacian operator and involving a subcritical nonlinear term which does not satisfy the technical Ambrosetti–Rabinowitz condition. By framing this problem in an appropriate variational setting, we prove a multiplicity theorem.

1 Introduction

In the standard framework of the p-Laplacian operator, there are a lot of interesting problems widely studied in the literature. A natural question is whether or not the results got in this classical context can be extended to the nonlocal case in the presence of fractional-type operators. Nonlocal fractional equations appear in many fields, and after the seminal works by Caffarelli and Silvestre [8–10], an increasing interest has been devoted to these topics; we refer, for instance, the recent papers [7, 12], the monograph [21] for several results on fractional variational problems and related topics, and [25–27] as well as references therein for some existence and multiplicity results involving the fractional p-Laplacian operator.

In most of the papers concerning with fractional Laplacian equation, it is assumed that the right-hand side has a superlinear, but subcritical growth (cf., e.g., [5, 22, 28, 29] and references therein). In the recent papers [3, 14] it has been firstly studied in the nonlocal setting the so-called asymptotically linear case: as it is well known, the lack of the classical Ambrosetti–Rabinowitz assumption requires more efforts in order to obtain a compactness Palais–Smale-type condition.

Further difficulties arise in the so-called resonant case (cf., e.g., [1] and references therein): indeed, the resonance affects both the compactness property and the geometry of the Euler–Lagrange functional arising in a suitable variational approach. For the local case we recall the contributions [2, 18, 24].

Motivated by this interest in the current literature, we would like to focus here our attention on p-fractional “asymptotically linear” problems, also in the presence of resonance. Indeed, we look for solutions of the nonlocal elliptic problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {(-\varDelta )^s_p u\ =\ \lambda |u|^{p-2}u+ f(x, u)} &{} \text{ in } \Omega ,\\ \displaystyle {u=0} &{} \text{ on } \mathbb {R} ^N{\setminus }\Omega ,\\ \end{array} \right. \end{aligned}$$

(1.1)

where $1<p<+\infty , s\in ]0,1[, (-\varDelta )^s_p$ denotes the fractional p-Laplacian which (up to normalization factors) may be defined for any $x\in {\mathbb {R}}^{N}$ as

$$\begin{aligned} (-\varDelta )^s_p\varphi (x)=2\lim _{\varepsilon \searrow 0^+}\int _{\mathbb {R} ^N{\setminus } B_{\varepsilon }(x)}\frac{\left| \varphi (x)-\varphi (y)\right| ^{p-2}(\varphi (x) -\varphi (y))}{|x-y|^{N+sp}}\mathrm {d}y \end{aligned}$$

along any $\varphi \in C^\infty _0(\mathbb {R} ^N)$, where $B_\varepsilon (x):=\left\{ y\in \mathbb {R} ^N\,:\,\left| x-y\right| <\varepsilon \right\}$, $\Omega$ is an open bounded domain of $\mathbb {R} ^N$ ($N>sp$) with Lipschitz boundary $\partial \Omega$ and $f:\Omega \times \mathbb {R} \rightarrow \mathbb {R}$ is a Carathéodory function such that:

$({h}_1)$ :

$\displaystyle \sup _{|t|\le a}|f(\cdot ,t)|\in L^\infty (\Omega ) \quad \forall \, a>0;$

$({h}_2)$ :

there exist

$$\begin{aligned} \lim _{|t|\rightarrow + \infty }\frac{f(x,t)}{|t|^{p-2}t}\ =\ 0 \end{aligned}$$

(1.2)

and

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{f(x,t)}{|t|^{p-2}t} \ = \alpha \in \mathbb {R}\ \end{aligned}$$

(1.3)

uniformly with respect to a.e. $x \in \Omega$.

Here we also deal with the resonant case, under the following further assumption (cf., e.g., [18]):

$(h_3)$ :

there exists

$$\begin{aligned} \lim _{|t|\rightarrow + \infty }\ \Big (f(x,t)t-pF(x,t)\Big )=\ +\infty \end{aligned}$$

uniformly with respect to a.e. $x \in \Omega$, where, as usual, we set

$$\begin{aligned} F(x,t):=\displaystyle \int _0^t f(x,\tau ) \; \mathrm{d}\tau . \end{aligned}$$

In order to state our multiplicity result we introduce some notations. Recalling that the Gagliardo seminorm of a measurable function $u:\mathbb {R} ^N\rightarrow \mathbb {R}$ is defined by

$$\begin{aligned}{}[u]_{s,p}:=\left( \int _{\mathbb {R} ^N\times \mathbb {R} ^N} \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\,\mathrm {d}x \mathrm {d}y\right) ^{\frac{1}{p}}, \end{aligned}$$

the fractional Sobolev space is defined by

$$\begin{aligned} W^{s,p}(\mathbb {R} ^N):=\{u\in L^p(\mathbb {R} ^N): [u]_{s,p}<+\infty \} \end{aligned}$$

and is equipped with the norm

$$\begin{aligned} \Vert u\Vert _{s,p}:=\left( |u|_p^p + [u]_{s,p}^p\right) ^{\frac{1}{p}}, \end{aligned}$$

where $|\cdot |_p$ denotes the norm on $L^p(\mathbb {R} ^N)$.

Our problem is set in the closed linear space

$$\begin{aligned} X(\Omega ):=\{u\in W^{s,p}(\mathbb {R} ^N): u(x)=0 \hbox { a.e. in } \mathbb {R} ^N{\setminus }\Omega \} \end{aligned}$$

endowed with the norm $\Vert \cdot \Vert = [\cdot ]_{s,p}$. The space $(X(\Omega ), \Vert \cdot \Vert )$ is uniformly convex and setting

$$\begin{aligned} p^*_s:=\displaystyle \frac{pN}{N-sp} \end{aligned}$$

it results:

$$\begin{aligned} X(\Omega )\hookrightarrow L^{\mu }(\Omega ) \hbox { continuously for }\mu \in [1,p^*_s] \end{aligned}$$

and

$$\begin{aligned} X(\Omega )\hookrightarrow \hookrightarrow L^{\mu }(\Omega )\hbox { compactly for }\mu \in [1,p^*_s[. \end{aligned}$$

(1.4)

Let us recall that $\lambda \in \mathbb {R}$ is an eigenvalue for $(-\varDelta )^s_p$ if there exists a nontrivial weak solution on $X(\Omega )$ of the problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {(-\varDelta )^s_p u\ =\ \lambda |u|^{p-2}u} &{} \text{ in } \Omega ,\\ \displaystyle {u=0} &{} \text{ on } \mathbb {R} ^N{\setminus }\Omega ,\\ \end{array} \right. \end{aligned}$$

(1.5)

that is $u\in X(\Omega ){\setminus }\{0\}$ such that

$$\begin{aligned} \begin{aligned}&\iint _{\mathbb {R} ^N\times \mathbb {R} ^N}\frac{\left| u(x)-u(y)\right| ^{p-2}(u(x)-u(y))(\varphi (x)-\varphi (y))}{\left| x-y\right| ^{N+sp}}\,\mathrm {d}x\mathrm {d}y\\&\quad -\lambda \int _{\Omega }\left| u(x)\right| ^{p-2}u(x)\varphi (x) \mathrm {d}x=0, \end{aligned} \end{aligned}$$

for every $\varphi \in X(\Omega )$.

As usual, the set of the eigenvalues is named spectrum and it is denoted by $\sigma ((-\varDelta )^s_p))$.

It is well known that if $p=2$ the spectrum of $(-\varDelta )^s$ in $X(\Omega )$ consists of a diverging sequence $\{\lambda _n\}_{n\in \mathbb {N}}$ of eigenvalues, repeated according to their multiplicity, satisfying $0<\lambda _1<\lambda _2\le \cdots \le \lambda _k\le \cdots$ (see [29]). Moreover, for $p=2$ the structure of $\sigma ((-\varDelta )^s)$ is very closed to that of the local operator $-\varDelta$ and also provides a decomposition by means of the eigenfunctions.

On the other hand, when $p\not =2$ the full spectrum of $(-\varDelta )^s_p$ is still almost unknown, even if some important properties of the first eigenvalue and of the higher-order (variational) eigenvalues have been established in [15, 19]. We also point out that a sequence of eigenvalues has been introduced in [17] by means of the cohomological index.

As we shall see in the proof of our main result, the definition of the quasi-eigenvalues proposed here and inspired to that in [11] fits in with our purposes, as a suitable decomposition of $X(\Omega )$ can be introduced and it turns out to be the known one for $p=2$ (cf. Sect. 2 for the details).

Thus, the interaction of the nonlinearity f with the spectrum $\sigma ((-\varDelta )^s_p)$ of the fractional p-Laplacian operator must be taken into account. To our knowledge the only contribution in this direction is by [17], where the authors obtain existence results via Morse theory.

Now we state our main result.

Theorem 1.1

Let $s\in ]0,1[, N>sp$ and $\Omega$ be an open bounded subset of $\mathbb {R} ^N$ with continuous boundary. Assume that $(h_1)$–$(h_2)$ hold, $f(x,\cdot )$ is odd for a.e. $x\in \Omega$ and that

$(h_4)$ :

there exist $h, k \in \mathbb {N}$ with $k\ge h$ such that

$$\begin{aligned} \alpha + \lambda<\ \beta _h\le \ \gamma _k\ <\lambda , \end{aligned}$$

where $\{\beta _n\}_{n\in \mathbb {N}}, \{\gamma _n\}_{n\in \mathbb {N}}$ are, respectively, as in (2.11) and (2.13) below.

Then, problem (1.1) has at least $k-h+1$ distinct pairs of nontrivial weak solutions, provided either

(a):

$\lambda \not \in \sigma ((-\varDelta )^s_p)$ or

(b):

$\lambda \in \sigma ((-\varDelta )^s_p)$ and $(h_3)$ holds true.

For the sake of completeness we recall here that a function $u\in X(\Omega )$ is a weak solution of problem (1.1) if

for every $\varphi \in X(\Omega )$.

This paper is organized as follows. In Sect. 2 we introduce the sequences of quasi-eigenvalues and depict some properties; then, in Sect. 3 we prove Theorem 1.1 by making use of a pseudo-index result recalled in Appendix as well as other classical tools.

2 Splitting of the fractional space $X(\Omega )$

In this section we adapt the arguments in [11, Section 5] to the nonlocal setting, thus constructing a first sequence of quasi-eigenvalues for $(-\varDelta )_p^s$ on $X(\Omega )$. Let us define the subset

$$\begin{aligned} {\mathcal S}:= \left\{ v \in X(\Omega ): \int _{\mathbb {R} ^N} |v(x)|^p \hbox {d}x = 1\right\} , \end{aligned}$$

(2.1)

the functional $\Phi :X(\Omega ) \rightarrow \mathbb {R}$ by

$$\begin{aligned} \Phi (u):= \Vert u\Vert ^p=[u]_{s,p}^p \end{aligned}$$

(2.2)

and

$$\begin{aligned} \beta _1 := \inf _{v \in {\mathcal S}}\Phi (v) > 0 \end{aligned}$$

($\beta _1$ is indeed the first eigenvalue of the p-fractional operator). Let us remark that

• $X(\Omega )$ is a reflexive Banach space;

• ${\mathcal S}$ is weakly closed in $X(\Omega )$ (by (1.4));

• $\Phi$ is coercive on $X(\Omega )$;

• $\Phi$ is weakly lower semicontinuous on $X(\Omega )$.

Then, by the generalized Weierstrass theorem, there exists a function $\psi _1 \in X(\Omega )$ such that

$$\begin{aligned} \int _{\mathbb {R} ^N} |\psi _1(x)|^p \mathrm {d}x = 1,\quad \Phi (\psi _1) = [\psi _1]_{s,p}^p =\beta _1; \end{aligned}$$

(2.3)

therefore,

$$\begin{aligned} \beta _1 |u|_p^p \le \Phi (u) \quad \hbox {for all}\, u\in X(\Omega ). \end{aligned}$$

Let us consider the linear operator ${\mathcal L}_1: L^p(\mathbb {R} ^N) \rightarrow \mathbb {R}$ related to $\psi _1$ ($\psi _1\in W^{s,p}(\mathbb {R} ^N)\Rightarrow \psi _1\in L^p(\mathbb {R} ^N)$ thus $|\psi _1|^{p-1}\in L^{p'}(\mathbb {R} ^N)$, where $p'$ is the conjugate of p) defined by

$$\begin{aligned} {\mathcal L}_1u := \int _{\mathbb {R} ^{N}} |\psi _1(x)|^{p-2}\psi _1(x)u(x)\,\mathrm {d}x. \end{aligned}$$

By definition, ${\mathcal L}_1 \in L^{p'}(\mathbb {R} ^N)$, while (2.3) implies ${\mathcal L}_1\psi _1 = 1$, then $\Vert {\mathcal L}_1\Vert _{L^{p'}} = 1$. Denoting again by ${\mathcal L}_1$ the restriction to the subspace $X(\Omega )$, it is also ${\mathcal L}_1 \in (X(\Omega ))'$.

Now, we define the new constraint

$$\begin{aligned} {\mathcal S}_1 := \{v \in {\mathcal S}: {\mathcal L}_1 v = 0\} = \ker ({\mathcal L}_1|_{\mathcal S}) \end{aligned}$$

and the corresponding constrained infimum

$$\begin{aligned} \beta _2 := \inf _{v \in {\mathcal S}_1}\Phi (v). \end{aligned}$$

We have that $\beta _2 > \beta _1$ (the first eigenvalue is isolated). We claim that also ${\mathcal S}_1$ is weakly closed in $X(\Omega )$. In fact, taking a sequence $\{v_m\}_{m\in \mathbb {N}} \subset {\mathcal S}_1$ and $v\in X(\Omega )$ such that

$$\begin{aligned} v_m \rightharpoonup v\; \hbox { weakly in}\,X(\Omega ), \end{aligned}$$

by (1.4) it follows that

$$\begin{aligned} v_m \rightarrow v\; \hbox { strongly in}\,L^p(\mathbb {R} ^N) \end{aligned}$$

and, since ${\mathcal L}_1 \in L^{p'}$ and $v_m \in {\mathcal S}_1$ for each $m \in \mathbb {N}$, we get that ${\displaystyle \int _{\mathbb {R} ^N} |v(x)|^p \mathrm {d}x = 1}$ and ${\mathcal L}_1 v_m \rightarrow {\mathcal L}_1 v$; therefore, $v \in {\mathcal S}_1$.

Thus, the generalized Weierstrass theorem applies again and there exists $\psi _2 \in {\mathcal S}_1$ such that $\Phi (\psi _2)=\beta _2$, i.e.,

$$\begin{aligned} \int _{\mathbb {R} ^N} |\psi _2(x)|^p \hbox {d}x = 1,\quad {\mathcal L}_1\psi _2=0,\quad \Phi (\psi _2)= \beta _2. \end{aligned}$$

The procedure can be repeated, so fixing any $n\in \mathbb {N}$ we can define some positive numbers

$$\begin{aligned} \beta _1 < \beta _2 \le \cdots \le \beta _n \end{aligned}$$

and some functions

$$\begin{aligned} \psi _1,\psi _2,\ldots ,\psi _n \in {\mathcal S}\end{aligned}$$

such that, for each $i \in \{1,\ldots ,n\}$, related to $\psi _i$ we can consider the linear operator ${\mathcal L}_{i}\in L^{p'}$ defined by

$$\begin{aligned} {\mathcal L}_{i}u := \int _{\mathbb {R} ^N} |\psi _{i}(x)|^{p-2}\psi _{i}(x)u(x)\mathrm {d}x \end{aligned}$$

(2.4)

such that

$$\begin{aligned}{}[\psi _i]_{s,p}^p = \beta _{i} \end{aligned}$$

(2.5)

and

$$\begin{aligned} {\mathcal L}_i\psi _i =\int _{\mathbb {R} ^N} |\psi _{i}(x)|^{p} \mathrm {d}x = 1, \end{aligned}$$

(2.6)

hence

$$\begin{aligned} \Vert {\mathcal L}_i\Vert _{L^{p'}} = 1, \end{aligned}$$

while ${\mathcal L}_j\psi _i=0$ for all $j\in \{1,\ldots , i-1\}$, thus

$$\begin{aligned} \psi _i \in \bigcap _{j=1}^{i-1}\ker ({\mathcal L}_j|_{\mathcal S}), \quad \hbox {if } i \ge 2. \end{aligned}$$

Therefore, we can define ${\mathcal S}_0:={\mathcal S}$,

$$\begin{aligned} {\mathcal S}_n := \{v \in {\mathcal S}: {\mathcal L}_1 v =\dots = {\mathcal L}_n v = 0\} = \bigcap _{i=1}^{n}\ker ({\mathcal L}_i|{\mathcal S}) \quad \hbox {if } n\in \mathbb {N}, \end{aligned}$$

and the corresponding constrained infimum

$$\begin{aligned} \beta _{n+1} := \inf _{v \in {\mathcal S}_n}\Phi (v) \quad \hbox {if } n \ge 0. \end{aligned}$$

(2.7)

We claim that there exists $\psi _{n+1} \in {\mathcal S}_n$ such that

$$\begin{aligned}{}[\psi _{n+1}]_{s,p}^p = \beta _{n+1}. \end{aligned}$$

(2.8)

To this aim, arguing as above, it is enough proving that each ${\mathcal S}_n$ is weakly closed. Indeed, if $\{v_m\}_{m\in \mathbb {N}}\subset {\mathcal S}_n$ weakly converges to $v\in X(\Omega )$, by (1.4) and ${\mathcal L}_i \in L^{p'}$, it follows that $|v|_p=1$ and ${\mathcal L}_i v_m \rightarrow {\mathcal L}_i v$ for all $i \in \{1,\ldots ,n\},$ hence $v \in {\mathcal S}_n$. Thus, $\Phi$ attains its infimum on ${\mathcal S}_n$ and (2.8) holds.

Summing up, by induction, we construct a sequence of positive numbers $\{\beta _n\}_{n\in \mathbb {N}}$, of functions $\{\psi _n\}_{n\in \mathbb {N}} \subset X(\Omega )$ and of linear operators $\{{\mathcal L}_n\}_{n\in \mathbb {N}} \subset L^{p'}$ such that (2.4)–(2.6) hold for all $n\in \mathbb {N}$; furthermore, it is

$$\begin{aligned} 0 < \beta _1 \le \beta _2 \le \cdots \le \beta _n\le \cdots \end{aligned}$$

and $\psi _n \ne \psi _m$ if $n \ne m$.

Now we recall that if $V\subseteq X$ is a closed subspace of a Banach space X, a subspace $W\subseteq X$ is a topological complement of V, briefly $X=V\oplus W$, if W is closed and every $x\in X$ can be uniquely written as $v+w$, with $v\in V$ and $w\in W$; furthermore, the projection operators onto V and W are linear and continuous; hence, there exists $L:=L(V,W)>0$ such that

$$\begin{aligned} \Vert v\Vert +\Vert w\Vert \le L\Vert v+w\Vert . \end{aligned}$$

(2.9)

When $X=V\oplus W$ and V has finite dimension, we say that W has finite codimension, with $\mathrm{codim}\, W=\dim V$.

By using again (1.4), the proofs of [11, Lemmas 5.2 and 5.3 and Proposition 5.4] can be adapted with minor changes to our setting; thus, the following properties can be stated:

• the increasing sequence $\{\beta _n\}_{n\in \mathbb {N}}$ diverges positively;

• fixing any $n \ge 1$ and setting

  $$\begin{aligned} X_n := & {} \mathrm{span}\{\psi _1,\ldots ,\psi _n\} = \left\{ v \in X(\Omega ): \; \exists \ b_1,\ldots ,b_n \in \mathbb {R}\ \hbox {s.t.}\ v = \sum _{i=1}^n b_i\psi _i\right\} , \\ Y_n := & {} \bigcap _{i=1}^{n}\ker ({\mathcal L}_i) = \{w \in X(\Omega ): {\mathcal L}_1 w =\dots = {\mathcal L}_{n}w = 0\}, \end{aligned}$$

  we have

  $$\begin{aligned} X(\Omega ) = X_n \oplus Y_n; \end{aligned}$$

  (2.10)

• taking $n \ge 1$, for all $w\in Y_n$ we get

  $$\begin{aligned} \beta _{n+1}\ \int _{\mathbb {R} ^N} |w(x)|^p \mathrm {d}x \le \ [w]_{s,p}^p; \end{aligned}$$

  (2.11)

• the sequence $\{\psi _n\}_{n\in \mathbb {N}}$ generates the whole space $X(\Omega )$.

Now, following [18] we introduce another sequence of positive numbers. For all $n\in {\mathbb {N}}$, taking $\psi _1$ as in (2.3), we set

$$\begin{aligned} {\mathbb {W} _n}=\{Z: Z \hbox { is a subspace of } X(\Omega ), \psi _1\in Z \hbox { and } \dim Z\ge n\} \end{aligned}$$

(2.12)

and

$$\begin{aligned} \gamma _n:=\inf _{Z\in {\mathbb {W} _n}}\sup _{u\in {\mathcal S}\cap Z}\Phi (u), \end{aligned}$$

(2.13)

with ${\mathcal S}$ as in (2.1). By the previous definitions, it follows that $\beta _1=\gamma _1$ and ${\mathbb {W} _{n+1}}\subseteq {\mathbb {W} _n}$; hence, $\{\gamma _n\}_{n\in \mathbb {N}}$ is an increasing sequence of quasi-eigenvalues.

Remark 2.1

For $p=2$ the sequences $\{\beta _n\}_{n\in \mathbb {N}}$ and $\{\gamma _n\}_{n\in \mathbb {N}}$ reduce to the known sequence of eigenvalues $\{\lambda _n\}_{n\in \mathbb {N}}$ of $(-\varDelta )^s$ (see, for instance, the paper [29]).

By using the genus we can construct a sequence of eigenvalues $\{\mu _n\}_{n\in \mathbb {N}}$ for the nonlinear operator $(-\varDelta )_s^p$ on $X(\Omega )$, alike in the case of local p-Laplacian as in [16, 20].

Let us consider the nonlinear eigenvalue problem (1.5) and set

$$\begin{aligned} \mu _n \,:=\, \inf _{A\in \Sigma _n}\sup _{u\in A{\setminus }\{0\}}\frac{\Phi (u)}{\displaystyle \int _{\mathbb {R} ^N} |u(x)|^p\; \mathrm{d}x},\quad n\in \mathbb {N} \end{aligned}$$

(2.14)

where (cf. Sect. 1) $\displaystyle \Sigma _n:=\{A\in \Sigma : \gamma (A)\ge n\}$ with $\Phi$ as in (2.2),

$$\begin{aligned} \Sigma :=\{A \subseteq X(\Omega ), \hbox {closed and symmetric w.r.t. the origin}\} \end{aligned}$$

(2.15)

and consider the even functional

$$\begin{aligned} \Psi (u)\ :=\ \frac{\Phi (u)}{\displaystyle \int _{\mathbb {R} ^N} |u(x)|^p\; \mathrm{d}x} \quad \hbox { on } X(\Omega ){\setminus }\{0\}. \end{aligned}$$

The critical values and the critical points of $\Psi$ restricted to the manifold ${\mathcal {S}}$ defined in (2.1) are eigenvalues and eigenfunctions of $(-\varDelta )^s_p$ on $X(\Omega )$, respectively. We can state the following proposition.

Proposition 2.2

For every $n\in \mathbb {N}$ the numbers $\mu _n$ in (2.14) are eigenvalues for the nonlinear operator $(-\varDelta )^s_p$ on $X(\Omega )$.

Proof

By [13, Lemma 4] and (1.4) the functional $\Psi \big |_{{\mathcal {S}}}$ satisfies the Palais–Smale condition. Then, by using a suitable version of the deformation lemma (cf., e.g., [6]), standard mini–max arguments give the result. $\square$

Furthermore, we point out that $\mu _1=\beta _1=\gamma _1$. Slight changes in the proof of [2, Proposition 2.9] and in [18, Remark 1.1(4)] provide the following proposition stating a comparison among the sequence $\{\mu _n\}_{n\in \mathbb {N}}$ and the sequences of quasi-eigenvalues $\{\beta _n\}_{n\in \mathbb {N}}$ and $\{\gamma _n\}_{n\in \mathbb {N}}$ of $(-\varDelta )^s_p$.

Proposition 2.3

For all $n\in \mathbb {N}$ we have that $\beta _n\le \mu _n\le \gamma _n$.

Remark 2.4

The properties of $\{\beta _n\}_{n\in \mathbb {N}}$ and Proposition 2.3 imply that $\{\mu _n\}_{n\in \mathbb {N}}$ and $\{\gamma _n\}_{n\in \mathbb {N}}$ are diverging sequences. Moreover, as $\{\gamma _n\}_{n\in \mathbb {N}}$ is increasing, we have also $\beta _h\le \gamma _k$ for $k\ge h\ge 1$; therefore, this inequality is not an assumption in $(h_4)$.

3 Proof of Theorem 1.1

From $(h_1)$ and (1.2) for all $\varepsilon >0$ there exists $K_\varepsilon >0$ such that

$$\begin{aligned} |f(x,t)|\le \varepsilon |t|^{p-1} + K_\varepsilon \quad \hbox { for a.e. } x\in \Omega \hbox { and for all } t\in \mathbb {R}. \end{aligned}$$

(3.1)

The weak solutions of problem (1.1) are the critical points of the $C^1$-functional

$$\begin{aligned} J_\lambda (u):=\frac{1}{p}\Vert u\Vert ^p - \frac{{\lambda }}{p} \int _\Omega |u(x)|^p \; \mathrm{d}x - \int _\Omega F(x,u(x))\mathrm{d}x \end{aligned}$$

(3.2)

on $X(\Omega )$ whose derivative is given by

for any $\varphi \in X(\Omega )$.

For the sake of simplicity we introduce the operator $A:X(\Omega )\rightarrow (X(\Omega ))^*$, defined for all $u,\varphi \in X(\Omega )$ by

$$\begin{aligned} \langle A(u),\varphi \rangle := \iint _{\mathbb {R} ^N\times \mathbb {R} ^N}\frac{\left| u(x)-u(y)\right| ^{p-2}(u(x)-u(y))(\varphi (x)-\varphi (y))}{\left| x-y\right| ^{N+sp}}\mathrm {d}x\mathrm {d}y. \end{aligned}$$

In next proposition we prove that the functional J satisfies the $(\mathrm C)$ condition (cf. Appendix) both in the nonresonant case and in the resonant one, up to assume also assumption $(h_3)$.

Proposition 3.1

Assume that $(h_1)$–$(h_2)$ hold. Then

• (i) if $\lambda \not \in \sigma ((-\varDelta )^s_p)$, the functional $J_\lambda$ in (3.2) satisfies $(\mathrm C)$ in $\mathbb {R};$

• (ii) if $\lambda \in \sigma ((-\varDelta )^s_p)$ and $(h_3)$ holds, the functional $J_\lambda$ in (3.2) satisfies $(\mathrm C)$ in $\mathbb {R}$.

Proof

(i) Let $c\in \mathbb {R}$ and $\{u_m\}_{m\in \mathbb {N}}$ be a sequence in $X(\Omega )$ such that (3.30) holds; then in particular

$$\begin{aligned} \begin{aligned} \langle A(u_m),\varphi \rangle&- \lambda \int _\Omega |u_m(x)|^{p-2} u_m(x)\varphi (x)\mathrm{d}x \\&-\int _\Omega f(x,u_m(x))\varphi (x)\mathrm{d}x = o(1), \end{aligned} \end{aligned}$$

(3.3)

for every $\varphi \in X(\Omega )$, where o(1) denotes an infinitesimal sequence.

In order to prove the statement, it is enough to show that $\{\Vert u_m\Vert \}_{m\in \mathbb {N}}$ is bounded (cf. [23, Proposition 1.3]). Then, arguing by contradiction, let us assume that

$$\begin{aligned} \Vert u_m\Vert \rightarrow +\infty \quad \hbox { as } m\rightarrow + \infty . \end{aligned}$$

(3.4)

Setting $w_m:=\displaystyle \frac{u_m}{\Vert u_m\Vert }$, $\{w_m\}_{m\in \mathbb {N}}$ is bounded in $X(\Omega )$ and there exists $w\in X(\Omega )$ such that, up to subsequences, we have

$$\begin{aligned} w_m\rightharpoonup w \quad \hbox { weakly in } X(\Omega ) \end{aligned}$$

(3.5)

and

$$\begin{aligned} w_m\rightarrow w \quad \hbox { strongly in } L^{p}(\Omega ). \end{aligned}$$

(3.6)

Evaluating (3.3) in $w_m-w$ and dividing by $\Vert u_m\Vert ^{p-1}$, we get

$$\begin{aligned} \langle A(w_m),w_m-w\rangle= & {} \lambda \int _\Omega |w_m(x)|^{p-2} w_m(x)\,(w_m-w)(x) \,\mathrm{d}x \, \nonumber \\&+ \int _\Omega \frac{f(x,u_m(x))}{\Vert u_m\Vert ^{p-1}}(w_m-w)(x)\;\mathrm{d}x + o(1). \end{aligned}$$

(3.7)

Let us analyze this last equation. Firstly, by (3.6) it follows that

$$\begin{aligned} \displaystyle {\left| \int _\Omega |w_m(x)|^{p-2} w_m(x) (w_m-w)(x)\mathrm{d}x\right| \le |w_m|_p^{p-1}|w_m-w|_p=o(1).} \end{aligned}$$

Furthermore, (3.1), (3.4) and (3.6) imply that

$$\begin{aligned} \left| \int _\Omega \frac{f(x,u_m(x))}{\Vert u_m\Vert ^{p-1}}(w_m-w)(x)\mathrm{d}x \right|\le & {} \varepsilon |w_m|_p^{p-1}|w_m-w|_p \nonumber \\&+\, \frac{K_\varepsilon }{\Vert u_m\Vert ^{p-1}}|w_m-w|_1 =o(1). \end{aligned}$$

(3.8)

Hence, by (3.7)

$$\begin{aligned} \langle A(w_m),w_m-w\rangle =o(1) \end{aligned}$$

and by [23, Proposition 1.3]

$$\begin{aligned} w_m\rightarrow w \quad \hbox { strongly in } X(\Omega ). \end{aligned}$$

(3.9)

Thus, by the definition of $w_m$ it follows $w\not =0$.

Now, dividing (3.3) by $\Vert u_m\Vert ^{p-1}$, for all $\varphi \in X(\Omega )$ we have that

$$\begin{aligned} \langle A(w_m),\varphi \rangle= & {} \lambda \int _\Omega |w_m(x)|^{p-2} w_m(x)\varphi (x)\mathrm{d}x \nonumber \\&+\,\int _\Omega \frac{f(x,u_m(x))}{\Vert u_m\Vert ^{p-1}}\varphi (x)\mathrm{d}x + o(1). \end{aligned}$$

(3.10)

Again (3.1), (3.4) and (3.6) give

$$\begin{aligned} \lim _{m\rightarrow + \infty }\int _\Omega \frac{f(x,u_m(x))}{\Vert u_m\Vert ^{p-1}}\varphi (x)\mathrm{d}x =0\,\,\,\hbox { for all } \varphi \in X(\Omega ). \end{aligned}$$

(3.11)

Therefore, by (3.9) and (3.11), passing to the limit in (3.10), we get

$$\begin{aligned} \langle A(w),\varphi \rangle = \lambda \int _\Omega |w(x)|^{p-2}w(x)\varphi (x)\mathrm{d}x \quad \hbox { for all } \varphi \in X(\Omega ). \end{aligned}$$

But this means that $\lambda \in \sigma ((-\varDelta )^s_p)$, against our assumption; thus, the proof is complete.

(ii) Let $c\in \mathbb {R}$ and $\{u_m\}_{m\in \mathbb {N}}$ be a sequence in $X(\Omega )$ such that (3.30) holds. Set

$$\begin{aligned} g(x,t):=\lambda t+f(x,t) \quad \hbox { for a.e. } x\in \Omega ,\,\, \forall \, t\in \mathbb {R}. \end{aligned}$$

(3.12)

By using (3.12) we have that

$$\begin{aligned} \frac{1}{p}\Vert u_m\Vert ^p - \int _{\Omega } G(x,u_m(x))\mathrm{d}x = c+ o(1) \end{aligned}$$

(3.13)

and

$$\begin{aligned} \Vert u_m\Vert ^p- \int _{\Omega } g(x,u_m(x))u_m(x)\mathrm{d}x = o(1), \end{aligned}$$

(3.14)

with $\displaystyle G(x,t):=\int _0^t g(x,\tau )\mathrm{d}\tau .$

By assumption $({h}_3)$ there exists $\eta _1>0$ such that

$$\begin{aligned} g(x,t)t-p\,G(x,t) \ge 0 \quad \quad \hbox {if}\,|t|\ge \eta _1, \hbox {for a.e.} x\in \Omega . \end{aligned}$$

(3.15)

On the other hand, by using condition $({h}_1)$ there exists $C_1=C_1(\eta _1)>0$ such that

$$\begin{aligned} \int _{\{|u_m|\le \eta _1\}} \left( g(x,u_m(x))u_m(x)-p\,G(x,u_m(x))\right) \mathrm{d}x \ge -C_1, \end{aligned}$$

(3.16)

for every $m\in \mathbb {N}$. Fixing $\varepsilon >0$, by (3.12) in addition to (1.2) of $({h}_2)$, there exists $\eta _\varepsilon >0$ such that

$$\begin{aligned} |g(x,t)|\le (|\lambda | + \varepsilon ) |t| \quad \quad \hbox {if}\, |t|>\eta _\varepsilon , \hbox {for a.e.} x\in \Omega . \end{aligned}$$

(3.17)

Now, taking $q\in ]p,p_s^*[$, there exists $C>0$ such that

$$\begin{aligned} |u|_{q}\le C^{\frac{1}{p}}\Vert u\Vert \quad \quad \hbox {for all } u\in X(\Omega ) \end{aligned}$$

(3.18)

(cf. (1.4)). Hence, let us set

$$\begin{aligned} \kappa := (2c + C_1)(2 (|\lambda | + \varepsilon )C)^{\frac{q}{q-p}}, \end{aligned}$$

(3.19)

with c as in (3.13) and $C_1$ as in (3.16).

Again by $({h}_3)$ we get the existence of $\eta _2 :=\eta _2(\kappa )>\max \{\eta _1, \eta _\varepsilon \}$ such that

$$\begin{aligned} g(x,t)t-p\,G(x,t) \ge \kappa \quad \quad \hbox {if}\,|t|\ge \eta _2, \hbox {for a.e.} x\in \Omega . \end{aligned}$$

(3.20)

Then, for $\eta _2$ as above, we define

$$\begin{aligned} A_m:=\{x\in \Omega : |u_m(x)|\ge \eta _2\} \end{aligned}$$

and

$$\begin{aligned} B_m :=\{x\in \Omega : |u_m(x)|\le \eta _2\}, \end{aligned}$$

for every $m\in \mathbb {N}$.

By (3.13)–(3.16) and (3.20) it follows that

$$\begin{aligned} \begin{aligned} pc + o(1)&= \int _{\Omega } \left( g(x,u_m(x))u_m(x)-p\,G(x,u_m(x))\right) \mathrm{d}x\\&= \int _{A_m} \left( g(x,u_m(x))u_m(x)-p\,G(x,u_m(x))\right) \mathrm{d}x\\&\quad + \int _{\{|u_m|\le \eta _1\}} \left( g(x,u_m(x))u_m(x)-p\,G(x,u_m(x))\right) \mathrm{d}x\\&\quad +\int _{\{\eta _1\le |u_m|\le \eta _2\}} \left( g(x,u_m(x))u_m(x)-p\,G(x,u_m(x))\right) \mathrm{d}x\\&\ge \ \kappa \, \mathrm{meas}(A_m) - C_1. \end{aligned} \end{aligned}$$

Hence, from the above inequality, one has

$$\begin{aligned} \mathrm{meas}(A_m)\le \frac{2c + C_1}{\kappa } + o(1) \quad \hbox {for all } m\in \mathbb {N}. \end{aligned}$$

(3.21)

Taking $r>p$, by (3.13) and (3.14) we have that

$$\begin{aligned} \begin{aligned} \left( \frac{1}{p} - \frac{1}{r}\right) \Vert u_m\Vert ^p&- \int _{\Omega } \left( G(x,u_m(x)) - \frac{1}{r} \, g(x,u_m(x))u_m(x)\right) \mathrm{d}x\\&= c + o(1). \end{aligned} \end{aligned}$$

(3.22)

Moreover, by $({h}_1)$ there exists $C_2:=C_2(\Omega ,g,\eta _2, r)>0$ such that

$$\begin{aligned} \left| \int _{B_m} \left( G(x,u_m(x)) - \frac{1}{r}\, g(x,u_m(x))u_m(x)\right) \;\mathrm{d}x \right| \le C_2,\,\,\,\forall m\in \mathbb {N}. \end{aligned}$$

(3.23)

Hence, by (3.22) and (3.23) we infer that

$$\begin{aligned} \begin{aligned} c + o(1)&\ge \left( \frac{1}{p} - \frac{1}{r}\right) \Vert u_m\Vert ^p\\&- \int _{A_m} \left( G(x,u_m(x)) - \frac{1}{r}\, g(x,u_m(x))u_m(x)\right) \mathrm{d}x - C_2. \end{aligned} \end{aligned}$$

Further, by (3.15) and (3.17) it follows that

$$\begin{aligned} \begin{aligned} c + o(1)&\ge \ \left( \frac{1}{p} - \frac{1}{r}\right) \Vert u_m\Vert ^p\\&- \int _{A_m} \left( \frac{1}{p} g(x,u_m(x))u_m(x) - \frac{1}{r}\, g(x,u_m(x))u_m(x)\right) \;\mathrm{d}x - C_2\\&\ge \ \left( \frac{1}{p} - \frac{1}{r}\right) \left( \Vert u_m\Vert ^p - \int _{A_m} (|\lambda | +\varepsilon )|u_m(x)|^p \;\mathrm{d}x\right) - C_2. \end{aligned} \end{aligned}$$

Now, by the Hölder inequality, (3.18), (3.19) and (3.21) we have that

$$\begin{aligned} \begin{aligned} c + o(1)&\ge \left( \frac{1}{p} - \frac{1}{r}\right) \left( \Vert u_m\Vert ^p - (\lambda +\varepsilon )\ |u_m|^p_{q}\ \mathrm{meas}(A_m)^{\frac{q-p}{q}}\right) - C_2\\&\ge \left( \frac{1}{p} - \frac{1}{r}\right) \Vert u_m\Vert ^p \left( 1 - (|\lambda | + \varepsilon )C\left( \frac{1}{(2 (\lambda + \varepsilon )C)^{\frac{q}{q-p}}} + o(1)\right) ^{\frac{q-p}{q}}\right) \\&-C_2. \end{aligned} \end{aligned}$$

Thus, the sequence $\{\Vert u_m\Vert \}_{m\in \mathbb {N}}$ is bounded in $X(\Omega )$. $\square$

Lemma 3.2

Assume that $(h_1)$–$(h_2)$ hold. Let $\beta _h$ be as in $(h_4)$ and $Y_{h-1}$ as in (2.10). Then, there exist $\rho >0$ and $c_0>0$ such that, setting $S_\rho :=\{u\in X(\Omega ): \Vert u\Vert = \rho \},$ the functional $J_\lambda$ in (3.2) verifies

$$\begin{aligned} J_\lambda (u)\ge c_0 \quad \hbox { for all } u\in S_\rho \cap Y_{h-1}. \end{aligned}$$

(3.24)

Proof

By $(h_2)$ it follows that, uniformly with respect to almost every $x \in \Omega$, there exist

$$\begin{aligned} \lim _{|t|\rightarrow +\infty }\frac{F(x,t)}{|t|^{p}} \ = 0 \end{aligned}$$

and

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{F(x,t)}{|t|^{p}} \ = \frac{\alpha }{p}. \end{aligned}$$

Therefore, for every $\varepsilon >0$ there exist $R_\varepsilon , \delta _\varepsilon >0$ such that, for almost every $x\in \Omega$,

$$\begin{aligned} |F(x,t)|\le \frac{\varepsilon }{p}|t|^p \quad \quad \hbox {if}\,|t|> R_\varepsilon \end{aligned}$$

(3.25)

and

$$\begin{aligned} \left| F(x,t)-\frac{\alpha }{p}|t|^p\right| \le \frac{\varepsilon }{p}|t|^p \quad \quad \hbox {if}\,|t|< \delta _\varepsilon , \end{aligned}$$

(3.26)

without loss of generality with $R_\varepsilon \ge 1$. On the other hand, by $(h_1)$, taking any $l\in [0, p_s^*-p[$, there exists $k_{R_\varepsilon }>0$ such that, for almost every $x\in \Omega$,

$$\begin{aligned} |F(x,t)| \le k_{R_\varepsilon }{|t|}^{l+p} \quad \quad \hbox {if}\,\delta _\varepsilon \le |t|\le R_\varepsilon . \end{aligned}$$

(3.27)

The inequalities (3.25)–(3.27) imply that for any $\varepsilon >0$ there exists $k_\varepsilon >0$ such that

$$\begin{aligned} F(x,t)\le \frac{\alpha + \varepsilon }{p}|t|^p + {k_\varepsilon }|t|^{l+p} \quad \quad \hbox {for a.e.} x\in \Omega , \hbox {for all}\,t\in \mathbb {R}. \end{aligned}$$

We infer that

$$\begin{aligned} \int _\Omega F(x,u(x))\mathrm{d}x\le \frac{\alpha + \varepsilon }{p}|u|^p_p + {k_\varepsilon }|u|^{l+p}_{l+p} \quad \hbox {for all } u\in X(\Omega ). \end{aligned}$$

For a suitable $k'_\varepsilon >0$ we have

$$\begin{aligned} J_\lambda (u)\ge \frac{1}{p}\Vert u\Vert ^p - \frac{\lambda + \alpha + \varepsilon }{p}|u|_p^p - k_\varepsilon ' \Vert u\Vert ^{l+p}\quad \hbox {for all } u\in X(\Omega ). \end{aligned}$$

(3.28)

Let us recall that by the decomposition (2.10) it is $X(\Omega )=X_{h-1}\oplus Y_{h-1}$, where $X_{h-1}:=\mathrm{span}\{\psi _1,\ldots ,\psi _{h-1}\}$ and $Y_{h-1}$ is its complement. Thus by (2.11) and (3.28) it follows that

$$\begin{aligned} J_\lambda (u)\ge \frac{1}{p} \left( 1 - \frac{\lambda + \alpha + \varepsilon }{\beta _{h}}\right) \Vert u\Vert ^p - k_\varepsilon ' \Vert u\Vert ^{l+p}\quad \hbox {for all}\,u\in Y_{h-1} \end{aligned}$$

and by $(h_4)$, for a suitable $\varepsilon$, there exists $k''_\varepsilon >0$ such that

$$\begin{aligned} J_\lambda (u)\ge k''_\varepsilon \Vert u\Vert ^p- k_\varepsilon '\Vert u\Vert ^{l+p} \quad \hbox {for all } u\in Y_{h-1}. \end{aligned}$$

Thus we conclude that if $\rho$ is small enough there exists $c_0>0$ such that (3.24) holds. $\square$

Lemma 3.3

Assume that $(h_1)$ and (1.2) hold. Let $\gamma _k$ as in $(h_4)$, $\mathbb {W} _{k}$ as in (2.12) and $c_0$ as in Lemma 3.2. Then, there exist a k-dimensional space $V\in \mathbb {W} _k$ and $c_\infty >c_0$ such that the functional $J_\lambda$ in (3.2) verifies

$$\begin{aligned} J_\lambda (u)\le c_\infty \quad \hbox { for all } u\in V. \end{aligned}$$

(3.29)

Proof

By (3.1), fixing any $\varepsilon >0$ there exists $C_\varepsilon >0$ such that

$$\begin{aligned} J_\lambda (u)\le \frac{1}{p}\Vert u\Vert ^p -\frac{\lambda }{p}|u|^p + \frac{\varepsilon }{2p}|u|^p_p + C_\varepsilon |u|_p \quad \hbox {for all } u\in X(\Omega ). \end{aligned}$$

Let $\gamma _k$ be as in $(h_4)$ and take $\varepsilon >0$ such that $\gamma _k+\varepsilon <\lambda$. From definition (2.13), for such a fixed $\varepsilon >0$ there exists a subspace $V^\varepsilon _k$ in $\mathbb {W} _k$, with $\dim V^\varepsilon _k\ge k$, such that

$$\begin{aligned} \gamma _k\le \sup _{u\in V^\varepsilon _k{\setminus }\{0\}}\frac{\Vert u\Vert ^p}{|u|^p}<\gamma _k+\frac{\varepsilon }{2}. \end{aligned}$$

Thus it results that

$$\begin{aligned} J_\lambda (u)\le \frac{1}{p}\left( \gamma _k+\varepsilon - \lambda \right) |u|^p + C_\varepsilon |u|_p \quad \hbox {for all } u\in V^\varepsilon _k \end{aligned}$$

and, as without loss of generality we can assume that $V^\varepsilon _k$ is a k-dimensional subspace, the functional $J_\lambda$ tends to $-\infty$ as $\Vert u\Vert$ diverges in $V^\varepsilon _k$, so there exists $c_\infty =c_\infty (\varepsilon )$ (with $c_\infty >c_0$), such that (3.29) holds. $\square$

Proof of Theorem 1.1

(a) Firstly, by Proposition 3.1—part (i) the functional $J_\lambda$ in (3.2) satisfies $(\mathrm C)$ in $\mathbb {R}$, and by assumption, it is even.

Let us consider $\beta _h$, $Y_{h-1}$, $\rho , c_0$ as in Lemma 3.2 and $\gamma _k, \mathbb {W} _{k}$, $V_k^\varepsilon$, $c_\infty$ as in Lemma 3.3.

Then, we consider the pseudo-index theory $(S_\rho \cap Y_{h-1}, {\mathcal {H}}^*, \gamma ^*)$ related to the genus and $S_\rho \cap Y_{h-1}$. By Remark 3.7 applied to $V:=V^\varepsilon _k$, $\partial B:= S_\rho$ and $W:=Y_{h-1}$, we get

$$\begin{aligned} \gamma \left( V^\varepsilon _k\cap h\left( S_\rho \cap Y_{h-1}\right) \right) \ge \dim V^\varepsilon _k - \text{ codim } Y_{h-1} \quad \hbox {for all } h\in {\mathcal {H}}^*, \end{aligned}$$

which implies

$$\begin{aligned} \gamma ^*(V^\varepsilon _k)\ge k-h+1. \end{aligned}$$

The proof is then complete: in fact Theorem 3.6 applies with ${\tilde{A}}:= V^\varepsilon _k$ and $S:=S_\rho \cap Y_{h-1}$ and J has at least $k-h+1$ distinct pairs of critical points corresponding to at most $k-h+1$ distinct critical values $c_i$, where $c_i$ is as in (3.32).

(b) In the resonant case, by Proposition 3.1—part (ii) the functional $J_\lambda$ satisfies $(\mathrm C)$, and we can proceed as above. $\square$

Remark 3.4

We point out that Theorem 1.1 holds also with slight changes in the proof when $(h_4)$ is replaced by

$$\begin{aligned} \lambda<\ \beta _h\le \ \gamma _k\ < \alpha + \lambda . \end{aligned}$$

Appendix: Abstract framework

Throughout this paper $(X,\Vert \cdot \Vert _X)$ is a Banach space, $(X',\Vert \cdot \Vert _{X'})$ its dual, I a $C^1$ functional on X, $I^b := \{u\in X : I(u)\le b\}$ the sublevel of I corresponding to $b\in \mathbb {R}$ and

$$\begin{aligned} K_c := \{u \in X:\ I(u) = c,\ \mathrm{d}I(u) = 0\} \end{aligned}$$

the set of the critical points of I in X at the critical level $c \in \mathbb {R}$.

In Sect. 3 we have seen that problem (1.1) has a variational structure; thus, next we recall some abstract tools used before.

Firstly, we recall the so-called Cerami’s variant of the Palais–Smale condition; even if it is a condition weaker than the classical one, it is enough in order to state a deformation theorem and some critical point theorems (cf. [1]).

Definition 3.5

The functional I satisfies the Cerami’s variant of the Palais–Smale condition at level c $\mathopen {(}c\in \mathbb {R} \mathclose {)}$, if any sequence $\{u_m\}_{m\in \mathbb {N}} \subseteq X$ such that

$$\begin{aligned} \{I(u_m)\}_{m\in \mathbb {N}} \hbox { is bounded and}\,\,\, \lim _{m \rightarrow +\infty }\Vert \mathrm{d}I(u_m)\Vert _{X'}(1 + \Vert u_m\Vert _X) = 0 \end{aligned}$$

(3.30)

converges in X, up to subsequences. In general, if $-\infty \le a<b\le +\infty$, I satisfies $(\mathrm C)$ in ]a, b[ if so is at each level $c\in ]a,b[$.

In the proof of our main theorem we use [1, Theorem 2.9] rewritten on Banach spaces where the index theory related to the genus acts. The proof is based on the use of a pseudo-index theory, and before introducing such a definition, we recall some notions of the index theory on Banach spaces X for an even functional with symmetry group $\mathbb {Z} _2 := \{\mathrm{id}, -\mathrm{id}\}$.

Define

$$\begin{aligned} \Sigma :=\ \{A \subseteq X \ \hbox {closed and symmetric w.r.t. the origin}\} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {H}} :=\{h\in C(X,X): h \text{ odd } \}. \end{aligned}$$

Taking $A \in \Sigma$, $A\ne \emptyset$, the genus of A is

$$\begin{aligned} \gamma (A) \ :=\ \inf \{k \in \mathbb {N}:\ \exists \psi \in C(A,\mathbb {R} ^k{\setminus }\{0\})\ \hbox {s.t.}\ \psi (-u) = - \psi (u) \text { for all } u\in A\}, \end{aligned}$$

if such an infimum exists, otherwise $\gamma (A) = +\infty$. Assume $\gamma (\emptyset ) = 0$.

The index theory $(\Sigma ,{\mathcal {H}}, \gamma )$ related to $\mathbb {Z} _2$ is also called genus (we refer for more details, e.g., to [30, Section II.5]).

According to [4], the pseudo-index related to the genus, an even functional $I:X\rightarrow \mathbb {R}$ and $S\in \Sigma$ is the triplet $(S, {\mathcal {H}}^*,\gamma ^*)$ such that ${\mathcal {H}}^*$ is a group of odd homeomorphisms and $\gamma ^*: \Sigma \longrightarrow \mathbb {N} \cup \{+\infty \}$ is the map defined by

$$\begin{aligned} \gamma ^*(A) :=\min _{h\in {\mathcal {H}}^*}\gamma (h(A)\cap S) \;\; \text { for all } A\in \Sigma . \end{aligned}$$

Since

$$\begin{aligned} \gamma (h(A)\cap S)=\gamma (A\cap h^{-1}(S)) \quad \hbox {for all } h\in {\mathcal {H}}^*, \end{aligned}$$

then

$$\begin{aligned} \gamma ^*(A) = \min _{h\in {\mathcal {H}}^*}\gamma (A\cap h(S)) \;\; \text { for all } A\in \Sigma . \end{aligned}$$

(3.31)

The following mini–max theorem was proved in [1, Theorem 2.9] in the setting of Hilbert spaces; the same proof holds on Banach spaces.

Theorem 3.6

Consider $a,b, c_0, c_\infty \in \bar{\mathbb {R}}$, $-\infty \le a<c_0<c_\infty <b\le +\infty$. Let I be an even functional, $(\Sigma ,{\mathcal {H}},\gamma )$ the genus theory on X, $S\in \Sigma$, $(S, {\mathcal {H}}^*,\gamma ^*)$ the pseudo-index theory related to the genus, I and S, with

$$\begin{aligned} {\mathcal {H}}^*=\{h\in {\mathcal {H}}: h \text{ bounded } \text{ homeomorphism } \text{ s.t. } h(u)=u \text{ if } u\not \in I^{-1}(]a,b[)\}. \end{aligned}$$

Assume that:

• (i) the functional I satisfies $(\mathrm C)$ in ]a, b[; 

• (ii) $S\subseteq I^{-1}([c_0,+\infty [);$

• (iii) there exist ${\tilde{k}}\in \mathbb {N}$ and ${\tilde{A}}\in \Sigma$ such that ${\tilde{A}}\subseteq I^{c_\infty }$ and $\gamma ^*({\tilde{A}})\ge {\tilde{k}}$.

Then the numbers

$$\begin{aligned} c_i:=\inf _{A\in \Sigma _i^*}\sup _{u\in A}I(u), \quad i\in \{1,\ldots , {\tilde{k}}\}, \end{aligned}$$

(3.32)

with $\Sigma _i^*:= \{A\in \Sigma : \gamma ^*(A)\ge i\}$, are critical values for I and

$$\begin{aligned} c_0\le c_1\le \cdots \le c_{\tilde{k}}\le c_\infty . \end{aligned}$$

Furthermore, if $c=c_i=\cdots =c_{i+r}$, with $i\ge 1$ and $i+r\le {\tilde{k}}$, then $\gamma (K_c)\ge r+1$.

Remark 3.7

In order to apply the theorem above, we need a lower bound for the pseudo-index of a suitable ${\tilde{A}}$ as in (iii) of Theorem 3.6. Thus, let us consider the genus theory $(\Sigma ,{\mathcal {H}},\gamma )$ on X and V, W two closed subspaces of X. If

$$\begin{aligned} \dim V<+\infty \quad \hbox { and } \quad \mathrm{codim}\, W<+\infty , \end{aligned}$$

then, for every odd bounded homeomorphism h on X and every open bounded symmetric neighborhood B of 0 in X, it results

$$\begin{aligned} \gamma (V\cap h(\partial B\cap W)) \ge \dim V - \mathrm{codim}\, W \end{aligned}$$

(cf. [1, Theorem A.2] and [2, Theorem 2.7]).