Nontrivial solutions to non-local problems with sublinear or superlinear nonlinearities

Abstract

Existence of nontrivial and multiple solutions for two types of non-local problems with sublinear or superlinear nonlinearities are investigated by linking theorems and index theory in critical point theory. Some results in the literature are extended.

1 Introduction

Fractional and non-local operators of elliptic type arise in a quite natural way in many different problems, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering,minimal surfaces, materials science, water waves and so on. The investigations of the problems involved these non-local operators are interesting and important from both pure mathematical research aspects and real-world applications, eg see [1, 2] and references therein.

Recently, variational methods and critical point theory have been proved to be powerful in dealing with these non-local elliptic problems after the paper [3] establishing the framework for the solvability of the following problems

$$\begin{aligned} \left\{ \begin{array}{l} (-\varDelta )^su=g(x,u) \ \text { in } \ \quad \varOmega , \\ u=0\ \quad \text { in } \ {{\mathbb {R}}^N} \backslash {\varOmega }{} , \end{array}\right. \end{aligned}$$

(1)

where $\varOmega \subset {{\mathbb {R}}}^N$ is an open bounded set with smooth boundary, $0<s<1$, $N > 2s$ and $(-\varDelta )^s$ is the fractional Laplace operator, which (up to normalization factors) may be defined as

$$\begin{aligned} (-\varDelta )^su(x)=\int _{{{\mathbb {R}}}^{N}} \frac{(u(x+y)+u(x-y)-2u(x))}{|y|^{N+2s}}dy, \quad \ x\in {\mathbb {R}}^{N}. \end{aligned}$$

They established the solvability of nontrivial solutions of the problems under the Ambrosetti- Rabinowitz superlinear condition for the nonlinearity g(x, u): there exist $\mu > 2$ and $r > 0$ such that for a.e. $x \in \varOmega ,\ t \in {{\mathbb {R}}}, \ |t|\ge r$, we have $0<\mu G (x,t)\le t g(x,t),$ where $G=\int _{0}^{t}g(x,\tau )d\tau$. When the nonlinearity g(x, u) satisfies a linear growth condition, solvability of the problem is studied in [4]. When g(x, u) is a lower order perturbation of the critical power, the classical Brezis-Nirenberg results are established in [5]. Some multiplicity results are also established either by Morse theory eg see [6, 7] or by fountain theorems eg see [8] where superlinear nonlinearities without Ambrosetti- Rabinowitz conditions are considered.

When g(x, u) is a linear perturbation such as $g(x,u)=\lambda q(x)u+f(x,u)$, where $\lambda =1$ or is a parameter related to the eigenvalues of certain eigenvalue problems and $q\in L^\infty (\varOmega )$, several researches considered the so-called non-resonance or resonant problems with Landesman-Lazer conditions when g(x, u) is bounded; for example, see [9, 10]. When the perturbation g(x, u) is superlinear and satisfies the Ambrosetti- Rabinowitz conditions, we refer to [11] for some results concerning the solvability of nontrivial solutions of the problem. In this paper, we first consider the case where the perturbation g(x, u) is sublinear and satisfies an extended form of Ahmad-Lazer-Paul type conditions; e.g. see [12,13,14] for some references. We remark that when f(x, u) is bounded, the Ahmad-Lazer-Paul type is more general than the Landesman-Lazer condition used in [10]; see [13, 15] and the references therein. Hence our results generalize the corresponding ones in [9, 10]. The other case we consider is the superlinear perturbation f(x, u), but we do not impose the standard Ambrosetti- Rabinowitz condition on it. This makes our results can be applied to more general nonlinearities. Moreover, not as in the literature we don’t need that the function q is bounded which makes us to deal with a different eigenvalue problem.

2 Main results

As in [3], we consider a more general non-local operator ${\mathcal {L}}_K$ with $(-\varDelta )^s$ as a special case, which is defined as follows:

$$\begin{aligned} {\mathcal {L}}_K u(x)=\int _{{{\mathbb {R}}}^{N}} \Big ((u(x+y)+u(x-y)-2u(x))\Big ){K}(y)dy, \quad \ x\in {\mathbb {R}}^{N}. \end{aligned}$$

Here $K: {\mathbb {R}}^{N}\backslash \{0\}\rightarrow (0,+\infty )$ is a function such that

$$\begin{aligned}&m K \in L^1({{\mathbb {R}}}^N),\quad \text {where } \ m(x)=\min \{|x|^2,1 \}; \end{aligned}$$

(2)

$$\begin{aligned}&\text {there}\ \text {exists}\ \theta >0 \ \text {such}\ \text {that}\ K(x)\ge \theta |x|^{-(N+2s)} \quad \text {for any}\ \ x\in {\mathbb R}^{N}\backslash \{0\}; \end{aligned}$$

(3)

$$\begin{aligned}&K(x)=K(-x)\quad \text {for any}\ x\in {{\mathbb {R}}}^{N}\backslash \{0\}. \end{aligned}$$

(4)

The function space X denotes the linear space of Lebesgue measurable functions from ${{\mathbb {R}}}^N$ to ${{\mathbb {R}}}$ such that the restriction to $\varOmega$ of any function g in X belongs to $L^2(\varOmega )$ and $(g(x)-g(y))\sqrt{K(x-y)}\in L^2({\mathbb {R}}^{2N}\backslash ({\mathcal {C}}\varOmega \times {\mathcal {C}}\varOmega ),dxdy)$, where ${\mathcal {C}}\varOmega ={{\mathbb {R}}}^{2N}\backslash \varOmega$.

Instead of (1), we consider the more general problem

$$\begin{aligned} \left\{ \begin{array}{l} -{\mathcal {L}}_K u=g(x,u) \quad \text { in } \ \varOmega , \\ u=0\quad \text { in } \ {{\mathbb {R}}^N} \backslash {\varOmega }{} . \end{array}\right. \end{aligned}$$

(5)

By a solution of (5), we mean a weak one. That is a $u\in X_0$ such that

$$\begin{aligned}&\int _{{\mathbb {R}}^{2N}} {(u(x)-u(y)) (\phi (x)-\phi (y))}{K(x-y)}dxdy =\int _{{\varOmega }}g(x,u(x))\phi (x) dx, \quad \forall \phi \in X_0, \end{aligned}$$

where the Hillbert space $X_0$ denotes

$$\begin{aligned} X_0:=\{ u\in X: u=0 \ \text {a.e.} \ \text {in} \ {{\mathbb {R}}^N} \backslash {\varOmega }{} \}, \end{aligned}$$

with the scalar product

$$\begin{aligned} \langle u,v \rangle =\int _{{\mathbb {R}}^{2N}} {(u(x)-u(y)) (v(x)-v(y))}{K(x-y)}dxdy \end{aligned}$$

and the norm $\left\| u\right\| ^2$ = $\langle u,u \rangle$.

The main results of the paper are existence theorems (Theorems 1–3 and Theorem 5) and a multiplicity result (Theorem 4) for two types equations driven by general non-local operators including fractional operators as special cases with sublinear or superlinear nonlinearities.

To be precise, in the first part of the paper we study the following problem with sublinear nonlinearities

$$\begin{aligned} \left\{ \begin{array}{l} -{\mathcal {L}}_Ku+a(x)u-\lambda _k u=g(x,u) \quad \text {in}\ \varOmega , \\ u=0\quad \text {in} \ {{\mathbb {R}}^N} \backslash {\varOmega }{} , \end{array}\right. \end{aligned}$$

(6)

where $\varOmega \subset {{\mathbb {R}}}^N$ is an open bounded domain with smooth boundary, $s\in (0,1),\ N> 2s,$ $a(x)\in L^{\frac{N}{2s}}(\varOmega )$, g(x, u) is not necessarily bounded and $\lambda _k$ is an eigenvalue of the problem

$$\begin{aligned} \left\{ \begin{array}{l} -{\mathcal {L}}_Ku+a(x)u=\lambda u \quad \text {in} \ \varOmega , \\ u=0\quad \text {in} \ {{{\mathbb {R}}}^N} \backslash {\varOmega }{} . \end{array}\right. \end{aligned}$$

(7)

We notice that since $a(x)\in L^{\frac{N}{2s}}(\varOmega )$ is not necessarily bounded, eigenvalue problem (7) does not seem to have been investigated in the literature. Hence we will first study problem (7). Particularly, we will prove that it has and only has a sequence of eigenvalues $\lambda _1<\lambda _2<\lambda _3<\cdots <\lambda _n<...$ with a finite multiplicity for each eigenvalue. The eigenspace corresponding to $\lambda _i$ is denoted by $E_i$. Suppose that $E_k=\text {span}\{\phi _1,\phi _2,\phi _3,...,\phi _m\}.$

We impose the following assumptions, where $G(x,t)=\int _{0}^{t}g(x,\tau )d\tau .$

($\hbox {g}_1$) $g\in C({\overline{\varOmega }}\times {\mathbb {R}},{{\mathbb {R}}}),\ |g(x,t)|\le C |t|^\alpha +C,$ for some $0\le \alpha <1$ and $C>0;$

($\hbox {G}_\pm$)$\frac{\int _{{\varOmega }}G\Big (x,\sum \nolimits _{i=1}^m\alpha _i\phi _i \Big )dx}{\left\| a\right\| ^{2\alpha }}\rightarrow \pm \infty ,$ as $\left\| a\right\| =\Big (\sum \limits _{i=1}^m\alpha _i^2\Big )^{\frac{1}{2}}\rightarrow \infty .$

If g(x, t) is odd for $t\in {\mathbb {R}},$ we can consider multiple solutions of (6) under some conditions. For example, we assume

($\hbox {g}_2$) there exists $r>0,$ s.t. $G(x,t)>0,$ as $(x,t)\in {\overline{\varOmega }}\times (0,r];$

($\hbox {g}_3$) $g(x,-t)=-g(x,t),$ as $(x,t)\in {\overline{\varOmega }} \times {{\mathbb {R}}} .$

We have the following results.

Theorem 1

Let $s\in (0,1),\ N>2s,\ \varOmega \subset {{\mathbb {R}}}^{N}$ be an open bounded domain with smooth boundary and let ${K}:\mathbb {R}^N \backslash \{0\} \rightarrow (0,+\infty )$ be a function satisfying (2), (3) and (4). Suppose that condition pair ($\hbox {g}_1$) ($\hbox {G}_+$) or ($\hbox {g}_1$) ($\hbox {G}_-$) holds, then (6) has at least one solution $u\in X_0.$

Theorem 2

Let $s\in (0,1),\ N>2s,\ \varOmega \subset {{\mathbb {R}}}^{N}$ be an open bounded domain with smooth boundary and let ${K}:\mathbb {R}^N \backslash \{0\} \rightarrow (0,+\infty )$ be a function satisfying (2), (3) and (4). Suppose that conditions ($\hbox {g}_1$) and ($\hbox {G}_+$) hold. If there exists $m\le k$ such that

$$\begin{aligned} \limsup _{t\rightarrow 0}\frac{g(x,t)}{t}<\lambda _m-\lambda _k \end{aligned}$$

(8)

and

$$\begin{aligned} \inf _{t\ne 0}\frac{g(x,t)}{t}\ge \lambda _{m-1}-\lambda _k \end{aligned}$$

(9)

uniformly for almost everywhere $x\in \varOmega ,$ then equation (6) has at least one nontrivial solution in $X_0.$

Theorem 3

Let $s\in (0,1),\ N>2s,\ \varOmega \subset {{\mathbb {R}}}^{N}$ be an open bounded domain with smooth boundary and let ${K}:\mathbb {R}^N \backslash \{0\} \rightarrow (0,+\infty )$ be a function satisfying (2), (3) and (4). Suppose that conditions ($\hbox {g}_1$) and ($\hbox {G}_-$) hold. If there exists $m\ge k$ such that

$$\begin{aligned} \liminf _{t\rightarrow 0}\frac{g(x,t)}{t}>\lambda _m-\lambda _k \end{aligned}$$

and

$$\begin{aligned} \sup _{t\ne 0}\frac{g(x,t)}{t}\le \lambda _{m+1}-\lambda _k \end{aligned}$$

uniformly for almost everywhere $x\in \varOmega ,$ then equation (6) has at least one nontrivial solution in $X_0.$

Theorem 4

Let $s\in (0,1),\ N>2s,\ \varOmega \subset {{\mathbb {R}}}^{N}$ be an open bounded domain with smooth boundary and let ${K}:\mathbb {R}^N \backslash \{0\} \rightarrow (0,+\infty )$ be a function satisfying (2), (3) and (4). Moreover, we assume that all the eigenfunctions of problem (7) belong to $L^\infty (\varOmega )$. If conditions ($\hbox {G}_-$) and ($\hbox {g}_1$-$\hbox {g}_3$) hold, then problem (6) has at least m solutions in $X_0,$ where m is the dimension of the eigenvalue $\lambda _k$.

Remark 1

The regularity assumption in Theorem 4 on the eigenfunctions of (7) is not too strong; e.g. see [2] for some related discussion for fractional problems.

Remark 2

Conditions ($\hbox {G}_\pm$), sometimes called generalized Ahmad-Lazer-Paul conditions, are now typical and widely used in the literature for dealing with elliptic PDEs on bounded domains or periodic solutions for Hamitonian systems with sublinear nonlinearities. But it seems that they haven’t appeared in non-local problems. As $\alpha =0$, ($\hbox {G}_\pm$) are reduced to the classical Ahmad-Lazer-Paul conditions, which include the well-known Landesman-Lazer conditions as special cases in variational problems, for dealing with similar problems but with bounded nonlinearities (e.g. see [14, 15] or Theorem 4.12 in [16]).

Extensions of the classical Ahmad-Lazer-Paul conditions to the present forms ($\hbox {G}_\pm$) for investigating unbounded problems were considered by a number of authors in early 1990s; e.g. see some references in [17]. The extensions to the present forms ($\hbox {G}_\pm$), either for elliptic PDEs or for periodic solutions for Hamiltonian systems, were also independently obtained in the first author’s Ph.D. thesis in 1992 under supervision of Professor Guo Dajun whom the paper dedicates to; e.g. see [12] and [18], where some further references are also available.

As in the literatures such as in [12] and the references therein, conditions ($\hbox {G}_\pm$) can be replaced by ones not involving the eigenfunctions of problem (7) (e.g. see [10]) if the following unique continuity property holds for the eigenfunctions of problem (7): all eigenfunctions corresponding to (7) have nodal set with zero Lebesgue measure, where the nodal set of a function $\phi$ in $\varOmega$ is the level set $\{x\in \varOmega , \phi (x)=0\}.$ Some information about the unique continuity property for non-local problems can be found in [10].

In the second part of the paper we study the following problem with superlinear nonlinearities

$$\begin{aligned} \left\{ \begin{array}{l} -{\mathcal {L}}_Ku+a(x)u=g(x,u) \quad \text {in} \ \varOmega , \\ u=0\quad \text {in} \ {{\mathbb {R}}^N} \backslash {\varOmega }{} , \end{array}\right. \end{aligned}$$

(10)

where K, $\varOmega$, $a(x)\in L^{\frac{N}{2s}}(\varOmega ),$ $s\in (0,1),\ N> 2s$ are all as in the first part.

We impose the following assumptions, where $G(x,t)=\int _{0}^{t}g(x,\tau )d\tau .$

($\hbox {a}_0$) $\lambda =0$ is not an eigenvalue of problem (7): $-{\mathcal {L}}_Ku+a(x)u=\lambda u,\ u\in X_0.$

($\hbox {g}_4$) $g\in C({\overline{\varOmega }}\times {\mathbb {R}},{{\mathbb {R}}})$, $G(x,t)\ge 0,\ g(x,t)=o(|t|)\ \text {as} \ t\rightarrow 0\ \text {uniformly} \ \text {in} \ {\bar{\varOmega }}.$

($\hbox {g}_5$) $\lim _{|t|\rightarrow +\infty }\frac{G(x,t)}{|t|^2}=+\infty \ \text {uniformly} \ \text {in}\ x\in {\bar{\varOmega }}.$

($\hbox {g}_6$) Set ${\widetilde{G}}(x,t)=\frac{1}{2}g(x.t)t-G(x,t).$ Then ${\widetilde{G}}(x,t)>0\ \text {for}\ t\ne 0$ and there is $r_0,c_0>0, \ \sigma >\max \{ 1,\frac{N}{2s}\}$ s.t.

$$\begin{aligned} |g(x,t)|^\sigma \le c_0{\widetilde{G}}(x,t)|t|^\sigma ,\ \text {if} \ |t|\ge r_0. \end{aligned}$$

Our main result reads as follows.

Theorem 5

Let $s\in (0,1),\ N>2s,\ \varOmega \subset {{\mathbb {R}}}^{N}$ be an open bounded domain with smooth boundary and let ${K}:\mathbb {R}^N \backslash \{0\} \rightarrow (0,+\infty )$ be a function satisfying (2), (3) and (4). Moreover, suppose that conditions ($\hbox {a}_0$), ($\hbox {g}_4$)–($\hbox {g}_6$) hold, then (10) has at least one nontrivial solution in $X_0.$

Remark 3

The conditions ($\hbox {g}_4$), ($\hbox {g}_5$),($\hbox {g}_6$) are more general than the Ambrosetti- Rabinowitz condition and a simple computation can prove that the superlinear function

$$\begin{aligned} g(x,t)=|u|^2\ln (1+|u|)-\frac{1}{2}|u|^2+|u|-\ln (1+|u|) \end{aligned}$$

satisfies ($\hbox {g}_4$) − ($\hbox {g}_6$) but does not satisfy Ambrosetti-Rabinowitz condition [19].

3 Preliminaries

In order to investigate eigenvalue problem (7), we first prove the following lemma.

Lemma 1

If $s\in (0,1),\ N>2s,\ \varOmega \subset {{\mathbb {R}}}^{2N}$ is an open domain, $a(x)\in L^{\frac{N}{2s}}(\varOmega )$ and $\ u_n\rightharpoonup u \in X_0,$ then

$$\begin{aligned} {\int _{\varOmega } a(x)|u_n(x)|^2dx} \rightarrow {\int _{\varOmega } a(x)|u(x)|^2dx} . \end{aligned}$$

Proof

Since $u_n\rightharpoonup u$ in $X_0,$ then by the imbedding theorem in [1, 3], $u_n^2$ is bounded in $L^{\frac{N}{N-2s}}(\varOmega ).$ So we may suppose

$$\begin{aligned} u_n^2\rightharpoonup u^2\quad \text {in} \ L^{\frac{N}{N-2s}}(\varOmega ). \end{aligned}$$

We assume that, up to a subsequence,

$$\begin{aligned} u_n^2(x)\rightarrow u^2(x)\quad \text {a.e.}\ x \in \varOmega . \end{aligned}$$

Then the proof is given by Vitali theorem. $\square$

Now we focus on the eigenvalue problem (7)

$$\begin{aligned} \left\{ \begin{array}{l} -{\mathcal {L}}_Ku+a(x)u=\lambda u \quad \text {in} \ \varOmega , \\ u=0\quad \text {in} \ {{\mathbb {R}}^N} \backslash {\varOmega }{} . \end{array}\right. \end{aligned}$$

Lemma 2

Let $s\in (0,1),\ N>2s,\ \varOmega \subset {{\mathbb {R}}}^{2N}$ be an open bounded domain and $a(x)\in L^{\frac{N}{2s}}(\varOmega )$.

• (a) Define

  $$\begin{aligned} \lambda _1:=\mathop {\inf } \limits _{u \in X_0,\Vert u\Vert _2=1}\frac{1}{2}\int _{{\mathbb {R}}^{2N}} {|u(x)-u(y)|^2}{K(x-y)}dxdy+{\int _{\varOmega } a(x)u(x)^2dx}. \end{aligned}$$

  Then $\lambda _1$ is finite and is a simple eigenvalue of (7) with a non-negative eigenfunction $e_1 \in X_0.$

• (b) The spectrum of problem (7) has and only has eigenvalues which can be listed $\lambda _1<\lambda _2\le \lambda _3\le \cdots \le \lambda _n\le ...$ and the corresponding eigenfunctions $\{e_k\}_{k\in N}$ form a base of Hilbert spaces $L^2(\varOmega )$ and $X_0.$

Proof

Denote $\widetilde{{\mathcal {J}}}(u)=\frac{1}{2}\int _{{\mathbb {R}}^{2N}} {|u(x)-u(y)|^2}{K(x-y)}dxdy+{\int _{\varOmega } a(x)u(x)^2dx}$, where $u\in X_0.$ By the Hölder and Sobolev inequalities, we have

$$\begin{aligned} \widetilde{{\mathcal {J}}}(u) \ge&\frac{1}{2}\left\| u\right\| ^2 -\varepsilon \left( \int _{\varOmega }|u|^{\frac{2N}{N-2s}}dx\right) ^\frac{N-2s}{N}-C(\varepsilon )\left( \int _{\varOmega }|a(x)|^{\frac{N}{2s}}dx \right) ^\frac{2s}{N} \\ \ge&\frac{1}{2} \left\| u\right\| ^2-C\varepsilon \left\| u\right\| ^2-C(\varepsilon ), \end{aligned}$$

for every $\varepsilon >0$. Hence, by fixing $\varepsilon$ small, $\widetilde{{\mathcal {J}}}(u)$ is coercive on $X_0$ and therefore is bounded below. Then $\lambda _1$ is a finite number. Let $u_n \in X_0,\Vert u_n\Vert _2=1$ be a minimizing sequence for $\widetilde{{\mathcal {J}}}$, that is, $\widetilde{{\mathcal {J}}}(u_n)\rightarrow \lambda _1.$ It is clear that $u_n$ is bounded in $X_0.$ Up to a subsequence suppose that $u_n\rightharpoonup u \ \text {in} \ X_0.$ By the compact imbedding $X_0\subset L^2(\varOmega )$ ([3]), we have $u_n\rightarrow u$ in $L^2(\varOmega ).$ Hence $\Vert u\Vert _2=1.$ By Lemma 1

$$\begin{aligned} {\int _{\varOmega } a(x)|u_n(x)|^2dx} \rightarrow {\int _{\varOmega } a(x)|u(x)|^2dx} . \end{aligned}$$

Hence by the lower semi-continuity of the norm in $X_0$

$$\begin{aligned} \widetilde{{\mathcal {J}}}(u)\le \liminf \limits _{n \rightarrow + \infty } \widetilde{{\mathcal {J}}}(u_n)=\lambda _1. \end{aligned}$$

Hence $\widetilde{{\mathcal {J}}}(u)=\lambda _1,\ \text {and} \ \lambda _1$ is achieved and is an eigenvalue of (7).

Since $u \in X_0$ implies that $|u| \in X_0$ and

$$\begin{aligned} ||u_n(x)|-|u_n(y)||^2\le |u_n(x)-u_n(y) |^2 \end{aligned}$$

we have that $\{|u_n|\}$ is also a minimizing sequence if $\{u_n\}$ is a minimizing sequence for $\widetilde{{\mathcal {J}}}$ in $X_0$. Therefore, there exists a non-negative eigenfunction $e_1$ corresponding to the first eigenvalue $\lambda _1$. In fact we can prove that every eigenfunction e corresponding to the first eigenvalue $\lambda _1$ doesn’t change sign. That is $e\ge 0$ or $e\le 0$ a.e. in $\varOmega .$ Obviously, $\widetilde{{\mathcal {J}}}(e)=\lambda _1= \widetilde{{\mathcal {J}}}(e_1).$ But, if $x\in \{e>0\}$ and $y\in \{e<0\}$, we have that

$$\begin{aligned} ||e(x)|-|e(y)||< |e(x)-e(y)|. \end{aligned}$$

This means that $\widetilde{{\mathcal {J}}}(|e|)<\widetilde{{\mathcal {J}}}(e),$ if both $\{e>0\}$ and $\{e<0\}$ have positive measure, which contradicts to $|e|\in X_0,\ \Vert e\Vert _2=1.$ As the proof of (c) in Proposition 9 ( [11]), we can get that $\lambda _1$ is simple.

For the proof of (b), we argue recursively. Assume that the claim holds for 1, ..., k and prove it for $k+1.$ By the definition of $\lambda _1,$ we have, just as Lemmas 2.14 and 2.15 in [20] for a similar problem,

$$\begin{aligned} \lambda _n=\mathop {\inf } \Big \{\Big (\left\| u\right\| ^2+\int _{\varOmega } a(x)u(x)^2dx\Big ):\Vert u\Vert _2=1, (u,e_1)=\cdots =(u,e_{n-1})=0\Big \}. \end{aligned}$$

So we get a sequence of eigenvalues

$$\begin{aligned} \lambda _1<\lambda _2\le \lambda _3\le \cdots \le \lambda _n\le ....\ \ \ \end{aligned}$$

From standard theory of compact linear operators, we know that the spectrum of problem (7) consists of just the above eigenvalues and the corresponding eigenfunctions form a base for both $X_0$ and $L^2(\varOmega )$. $\square$

We can also characterize the eigenvalues by subspaces of $X_0$.

Proposition 1

We have

$$\begin{aligned} \lambda _n=\mathop {\max }\limits _{V \in V_{n-1}}\mathop {\inf } \limits _{u \in V^\perp ,\Vert u\Vert _2=1} \Big (\left\| u\right\| ^2+\int _{\varOmega } a(x)u(x)^2dx\Big ) \end{aligned}$$

where $V_{n-1}$ is the set of $n-1$ dimensional subspaces in $X_0$.

Proof

Denote

$$\begin{aligned} {\widetilde{\lambda }}_n:=\mathop {\max }\limits _{V \in V_{n-1}}\mathop {\inf } \limits _{u \in V^\perp ,\Vert u\Vert _2=1} \Big (\left\| u\right\| ^2+\int _{\varOmega } a(x)u(x)^2dx\Big ). \end{aligned}$$

By the variational definition of $\lambda _n,\ {\widetilde{\lambda }}_n\ge \lambda _n.$ On the other hand, for any $V \in V_{n-1},$ there exists $u (u\ne 0),$ $u=\sum \limits _{i=1}^nx_je_j\in V^\perp$ where $\{e_i\}_{1\le i \le n}$ are the corresponding eigenfunctions of $\{\lambda _i\}_{1\le i \le n}$ such that

$$\begin{aligned} \left\| u\right\| ^2+\int _{\varOmega } a(x)u^2(x)dx=\sum \limits _{i=1}^n\lambda _jx_j^2\int _{\varOmega } e_j^2dx\le \lambda _n\int _{\varOmega } u^2dx. \end{aligned}$$

(11)

Hence

$$\begin{aligned} \mathop {\inf } \limits _{u \in V^\perp ,\Vert u\Vert _2=1} \Big (\left\| u\right\| ^2+{\int _{\varOmega } a(x)u^2(x)dx}\Big )\le \lambda _n . \end{aligned}$$

So, $\ {\widetilde{\lambda }}_n\le \lambda _n.$ The proposition is proved. $\square$

Lemma 3

If the eigenvalues of (7) are listed as follows $\lambda _1<\lambda _2\le ...\le \lambda _k\le 0< \lambda _{k+1}\le ...$ and

$$\begin{aligned} Y= & {} \{e_1,...,e_k\} \ \text {where} \ \{e_k\} \ \text {are the corresponding eigenfunctions}\\ Z= & {} \{u\in X_0, \langle u,v\rangle _{L^2}=0 ,\forall v \in Y\}, \end{aligned}$$

then we have

$$\begin{aligned} \delta =\mathop {\text {inf}} \limits _{u \in Z,\left\| u\right\| =1} \left\| u\right\| ^2+{\int _{\varOmega } a(x)u^2(x)dx}>0. \end{aligned}$$

Proof

By the above variational characterization of $\lambda _{k+1}$,

$$\begin{aligned} \left\| u\right\| ^2+\int _{\varOmega } a(x)u^2(x)dx\ge \lambda _{k+1}\int _{\varOmega }u^2, \ \text {for} \ \text {any} \ u\in Z. \end{aligned}$$

If the lemma were not true, there exist $u_n\in Z, n\in {\mathbb {N}}$

$$\begin{aligned} \left\| u_n\right\| ^2+\int _{\varOmega } a(x)u_n^2(x)dx<\frac{1}{n}\left\| u_n\right\| ^2. \end{aligned}$$

Set $v_n=\frac{u_n}{\left\| u_n\right\| }.$ Then

$$\begin{aligned} 1+\int _{\varOmega } a(x)v_n^2(x)dx\le \frac{1}{n}. \end{aligned}$$

Without loss of generality, assume $v_n\rightharpoonup v \ \text {in} \ X_0$.

By Lemma 1

$$\begin{aligned} {\int _{\varOmega } a(x)v_n^2(x)dx} \rightarrow {\int _{\varOmega } a(x)v^2(x)dx} . \end{aligned}$$

Hence,

$$\begin{aligned} 1+\int _{\varOmega } a(x)v^2(x)dx\le 0, \end{aligned}$$

(12)

which implies that $v\ne 0.$ Since $v\in Z,$

$$\begin{aligned} \left\| v\right\| ^2+\int _{\varOmega } a(x)v^2(x)dx\ge \lambda _{k+1}\int _{\varOmega } v_n^2(x)dx. \end{aligned}$$

(13)

We change (12) to

$$\begin{aligned} \left\| v\right\| ^2+\int _{\varOmega } a(x)|v(x)|^2dx\le \left\| v\right\| ^2-1. \end{aligned}$$

(14)

By combining (13) and (14), we have

$$\begin{aligned} \left\| v\right\| ^2\ge 1+\lambda _{k+1}\int _{\varOmega }v^2(x)dx>1. \end{aligned}$$

But,

$$\begin{aligned} \left\| v\right\| ^2\le \lim _{n \rightarrow +\infty }\left\| v_n\right\| ^2=1. \end{aligned}$$

The contraction ends the proof. $\square$

The proofs of the existence of one (nontrivial) solution rely on the standard linking theorems (e.g. see [16, 20]) and the proof of Theorem 4 needs the following proposition from ${\mathbb {Z}}^2$-index theory (e.g. see [21, 22]).

Proposition 2

Let ${\mathcal {J}}$ be an even and $C^1-functional$ on a Banach space X,  satisfying P-S condition and $f(\theta )=0.$ If there is $\rho >0$ and a $m-$dimensional subspace $X_1$ s.t.

$$\begin{aligned} \sup _{x\in X_1 \cap S(\theta ,\rho )}{\mathcal {J}}(x)<0; \end{aligned}$$

and there is another j-dimensional subspace $X_2 (j<m)$ s.t.

$$\begin{aligned} \inf _{x\in X_2^\perp }{\mathcal {J}}(x)>-\infty \end{aligned}$$

where $X_2^\perp$ is the direct compliment subspace of $X_2,$ then ${\mathcal {J}}$ has at least $m-j$ pair critical points.

4 Proof of main results

Due to the variational nature of the problem, in order to find weak solutions for problem (6), in the following we will look for critical points of the functional ${\mathcal {J}}$ on $X_0$

$$\begin{aligned} {\mathcal {J}}(u)= \frac{1}{2}\left\| u\right\| ^2 +\frac{1}{2}\int _{\varOmega } a(x){(u(x))}^2dx-\frac{1}{2}\int _{\varOmega } \lambda _k{(u(x))}^2dx -\int _{\varOmega }G(x,u)dx, \end{aligned}$$

where $\lambda _k$ is an eigenvalue of (7). Here all eigenvalues of (7) are listed as $\lambda _1<\lambda _2<\lambda _3<\cdots<\lambda _n<\ldots$ and the corresponding eigenspaces are denoted by $E_k$ ($k=1,2,\ldots$).

Moreover, it is known that under the conditions of our theorems ${\mathcal {J}}\in C^1(X_0,{\mathbb {R}})$ and for any $u,\ \phi \in X_0$,

$$\begin{aligned} \langle {\mathcal {J}}'(u),\phi \rangle&= \int _{\mathbb R^{2N}}(u(x)-u(y))(\phi (x)-\phi (y)){K(x-y)}dxdy+\int _{\varOmega } a(x){u(x)}\phi (x) dx\\&\quad -\int _{\varOmega } \lambda _k{u(x)}\phi (x) dx-\int _{\varOmega }g(x,u(x))\phi (x) dx. \end{aligned}$$

Let us write

$$\begin{aligned} u={\bar{u}} + u^0 +{\widetilde{u}},\ u\in X_0, \end{aligned}$$

where

$$\begin{aligned} {\bar{u}} \in \sum \limits _{i< k}E_i , \ \ u^0\in E_k\ \text {and}\ {\widetilde{u}}\in \overline{\sum \limits _{i\ge {k+1}}E_i}. \end{aligned}$$

Lemma 4

Under condition pair ($\hbox {g}_1$), ($\hbox {G}_+$) or ($\hbox {g}_1$), ($\hbox {G}_-$), the functional ${\mathcal {J}}$ defined above satisfies P-S condition on $X_0.$

Proof

We only prove the case where ($\hbox {g}_1$) and ($\hbox {G}_+$) hold. The other case can be proved similarly.

Suppose that $(u_n)\in X_0$ satisfies

$$\begin{aligned} {\mathcal {J}}'(u_n) \rightarrow 0,\ \text {as} \, n\rightarrow +\infty , \end{aligned}$$

and

$$\begin{aligned} |{\mathcal {J}}(u_n)| \le C. \end{aligned}$$

We have, noticing a similar inequality in (11)

$$\begin{aligned}&\langle {\mathcal {J}}'(u_n),-{\bar{u}}_n\rangle \\&\quad = -\int _{{\mathbb {R}}^{2N}}|{\bar{u}}_n(x)-\bar{u}_n(y)|^2{{K(x-y)}}dxdy-\int _{\varOmega } a(x)|{\bar{u}}_n(x)|^2 dx+\int _{\varOmega } \lambda _k|{\bar{u}}_n(x)|^2 dx\\&\qquad + \int _{\varOmega }g(x,u_n(x)){\bar{u}}_n(x) dx\\&\quad \ge (\lambda _k-\lambda _{k-1})\int _{\varOmega } |{\bar{u}}_n(x)|^2 dx-\int _{\varOmega } |{\bar{u}}_n(x)|\Big (C|{\bar{u}}(x) + u^0(x) +\widetilde{u}_n(x)|^\alpha +C\Big ) dx\\&\quad \ge (\lambda _k-\lambda _{k-1})\int _{\varOmega } |{\bar{u}}_n|^2 dx-C\int _{\varOmega } |{\bar{u}}_n| dx-C\int _{\varOmega }|{\bar{u}}_n|(|\bar{u}_n|^\alpha +| u_n^0|^\alpha +|{\widetilde{u}}_n|^\alpha )dx\\&\quad \ge (\lambda _k-\lambda _{k-1}-\varepsilon )\int _{\varOmega } |\bar{u}_n|^2 dx-C\int _{\varOmega }|{\bar{u}}_n||u_n^0|^\alpha dx-C\int _{\varOmega }|{\bar{u}}_n||{\widetilde{u}}_n|^\alpha dx-C(\varepsilon ), \end{aligned}$$

where $C(\varepsilon ) > 0$ is a universal constant dependent on the arbitrary $\varepsilon > 0.$ Fixing $\varepsilon >0$ sufficiently small and noticing that $\sum \nolimits _{i\le {k-1}}E_i$ is finite dimensional, we have

$$\begin{aligned} \left\| {\bar{u}}_n\right\| ^2\le C\left\| \widetilde{u}_n\right\| ^{2\alpha }_{2\alpha } + C\left\| u_n^0 \right\| ^{2\alpha }_{2\alpha }+ C. \end{aligned}$$

(15)

By a similar computation and noticing Lemma 3, we have

$$\begin{aligned}& \langle {\mathcal {J}}'(u_n),{\widetilde{u}}_n\rangle \\& \quad =\int _{{\mathbb {R}}^{2N}}{|{\widetilde{u}}_n(x)-\widetilde{u}_n(y)|^2}{K(x-y)}dxdy+\int _{\varOmega }a(x)|{\widetilde{u}}_n(x)|^2 -\int _{\varOmega }\lambda _k|{\widetilde{u}}_n(x)|^2 \\& \qquad + \int _{\varOmega }g(x,u_n(x)){\widetilde{u}}_n(x) dx\\&\quad \ge \int _{{\mathbb {R}}}^{2N}{|{\widetilde{u}}_n(x)-\widetilde{u}_n(y)|^2}{K(x-y)}dxdy+\int _{\varOmega }a(x)|\widetilde{u}_n(x)|^2dx+\int _{\varOmega }g(x,u_n(x)){\widetilde{u}}_n(x) dx\\&\qquad -\frac{\lambda _k}{\lambda _{k+1}}\Big [\int _{{\mathbb R^{2N}}}{|{\widetilde{u}}_n(x)-\widetilde{u}_n(y)|^2}{K(x-y)}dxdy+\int _{\varOmega }a(x)|{\widetilde{u}}_n(x)|^2 dx\Big ]\\&\quad = \Big (1-\frac{\lambda _k}{\lambda _{k+1}}\Big ) \Big [\int _{{\mathbb {R}}}^{2N}{|{\widetilde{u}}_n(x)-\widetilde{u}_n(y)|^2}{K(x-y)}dxdy+\int _{\varOmega }a(x)|{\widetilde{u}}_n(x)|^2 dx\Big ]\\&\qquad +\int _{\varOmega }g(x,u_n(x)){\widetilde{u}}_n(x) dx\\&\quad \ge \Big (1-\frac{\lambda _k}{\lambda _{k+1}}\Big ) \Big [\int _{{\mathbb {R}}^{2N}}{|{\widetilde{u}}_n(x)-\widetilde{u}_n(y)|^2}{K(x-y)}dxdy\\&\qquad +\int _{\varOmega }a(x)|{\widetilde{u}}_n(x)|^2 \Big ]-\int _{\varOmega }(C|u_n|^\alpha +C)|{\widetilde{u}}_n(x)| dx\\&\quad \ge \delta \Big (1-\frac{\lambda _k}{\lambda _{k+1}}\Big ) \int _{{\mathbb {R}}^{2N}}{|{\widetilde{u}}_n(x)-\widetilde{u}_n(y)|^2}{K(x-y)}dxdy\\&\qquad -C\int _{\varOmega }|{\widetilde{u}}_n|(C|\widetilde{u}_n +{\bar{u}}_n+u_n^0|^\alpha +C)dx\\&\quad \ge \delta \Big (1-\frac{\lambda _k}{\lambda _{k+1}}\Big )\left\| \widetilde{u}_n\right\| ^2-C\int _{\varOmega }|{\widetilde{u}}_n| dx -C\int _{\varOmega }|{\widetilde{u}}_n|(|{\bar{u}}_n|^\alpha +| u_n^0|^\alpha +|{\widetilde{u}}_n|^\alpha )dx\\&\quad \ge \delta \Big (1-\frac{\lambda _k}{\lambda _{k+1}}-\varepsilon \Big )\left\| \widetilde{u}_n\right\| ^2-C(\varepsilon )\int _{\varOmega }|{\bar{u}}_n|^{2\alpha } dx-C(\varepsilon )\int _{\varOmega }|\bar{u}_n^0|^{2\alpha }dx-C(\varepsilon ), \end{aligned}$$

where $C(\varepsilon ) > 0$ is a universal constant dependent on the arbitrary $\varepsilon > 0.$ Similarly, noticing that $\langle {\mathcal {J}}'(u_n),{\widetilde{u}}_n\rangle \le o(1)\left\| {\widetilde{u}}_n\right\| ,$ and fixing $\varepsilon >0$ sufficiently small, we have

$$\begin{aligned} \left\| {\widetilde{u}}_n\right\| ^2\le C\left\| \bar{u}_n\right\| ^{2\alpha }_{2\alpha } + C\left\| u_n^0 \right\| ^{2\alpha }_{2\alpha }+ C. \end{aligned}$$

(16)

By enlarging the term $\Vert {\widetilde{u}}_n\Vert ^{2\alpha }_{2\alpha }$ in the right side of (15) to $C\Vert {\widetilde{u}}_n\Vert ^{2\alpha }$ by embedding inequality and inserting the inequality (16) to (15) and then using Young inequality, we can get the following inequality

$$\begin{aligned} \left\| {\bar{u}}_n\right\| ^2\le C\left\| u_n^0 \right\| ^{2\alpha }_{2\alpha }+ C. \end{aligned}$$

(17)

Similarly, we have

$$\begin{aligned} \left\| {\widetilde{u}}_n\right\| ^2\le C\left\| u_n^0 \right\| ^{2\alpha }_{2\alpha }+ C. \end{aligned}$$

(18)

By

$$\begin{aligned}&\Big |\int _{\varOmega }\Big (G(x,u_n)-G(x,u_n^0)\Big ) dx\Big |=|\int _{\varOmega }dx\int _0^1g(x,u_n^0+s({\widetilde{u}}_n+\bar{u}_n))(({\widetilde{u}}_n+{\bar{u}}_n))ds| \\&\quad \le \int _\varOmega dx\int _0^1 (|\widetilde{u}_n|+ |\bar{u}_n |)(C|u_n^0+s( {\widetilde{u}}_n+{\bar{u}}_n)|^\alpha +b)ds \\&\quad \le C\int _\varOmega (|{\widetilde{u}}_n||u_n^0|^\alpha + |{\widetilde{u}}_n|^{1+\alpha } + |{\widetilde{u}}_n||{\bar{u}}_n |^\alpha +b |{\widetilde{u}}_n|)dx\\&\qquad + \int _\varOmega ( |{\bar{u}}_n| |u_n^0|^\alpha + |{\bar{u}}_n ||{\widetilde{u}}_n|^{\alpha }+ |{\bar{u}}_n |^{1+\alpha }+b |{\bar{u}}_n |)dx, \end{aligned}$$

and estimating the above each term with Hölder and embedding inequalities and (17) and (18), we can get

$$\begin{aligned} \Big |\int _{\varOmega }\Big (G(x,u_n)-G(x,u_n^0)\Big ) dx\Big |\le C\left\| u_n^0 \right\| ^{2\alpha }_{2\alpha }+ C. \end{aligned}$$

(19)

By $|{\mathcal {J}}(u_n)| \le C$ and the inequality

$$\begin{aligned} \int _{{\mathbb {R}}^{2N}}{|{\bar{u}}_n(x)-\bar{u}_n(y)|^2}{K(x-y)}dxdy+\int _{\varOmega }a(x)|{\bar{u}}_n(x)|^2 dx-\int _{\varOmega }\lambda _k|{\bar{u}}_n(x)|^2 dx\le 0, \end{aligned}$$

we have

$$\begin{aligned}&-C\le \frac{1}{2}\int _{{\mathbb {R}}^{2N}}{|\widetilde{u}_n(x)-\widetilde{u}_n(y)|^2}{K(x-y)}dxdy+\frac{1}{2}\int _{\varOmega }a(x)|\widetilde{u}_n(x)|^2 \\&\quad -\int _{\varOmega }\lambda _k|{\widetilde{u}}_n(x)|^2 -\int _{\varOmega }G(x,u_n) dx. \end{aligned}$$

Noticing

$$\begin{aligned} \int _{\varOmega }a(x)|{\widetilde{u}}_n(x)|^2 dx\le \int _{\varOmega }|a(x)| |{\widetilde{u}}_n(x)|^2 dx\le \Vert a\Vert _{\frac{N}{2s}}\Vert \widetilde{u}_n\Vert ^2_{\frac{2N}{N-2s}}\le C\left\| {\widetilde{u}}_n\right\| ^2 \end{aligned}$$

and a similar inequality $-\int _{\varOmega }\lambda _k|\widetilde{u}_n(x)|^2 dx\le C\left\| {\widetilde{u}}_n\right\| ^2$, we change the above inequality to

$$\begin{aligned} -C\le C\left\| \widetilde{u}_n\right\| ^2-\int _{\varOmega }\Big [G(x,u_n)-G(x,{u_n}^0)\Big ] dx-\int _{\varOmega }G(x,{u_n}^0) dx. \end{aligned}$$

Moreover, by (18) and (19), we have

$$\begin{aligned} -C\le C\left\| u_n^0 \right\| ^{2\alpha }_{2\alpha }+ C-\int _{\varOmega }G(x,{u_n}^0) dx. \end{aligned}$$

(20)

Hence, $\{u_n^0\}$ is bounded by ($\hbox {G}_+$). Therefore, by (17) and (18), $\{u_n\}$ is bounded in $X_0$. A standard argument ( [3]) implies that ${\mathcal {J}}$ satisfies Palais-Smale condition on $X_0.$ $\square$

Proof of Theorem 2

Write $X_0=\sum \nolimits _{i\le m-1}E_i\oplus \overline{\sum \nolimits _{i\ge m}E_i}.$ We claim that

• (i) $\exists \ \rho ,\ d>0$ s.t. ${\mathcal {J}}\ge d$ on $\Big \{ u\in \sum \nolimits _{i\ge m}E_i|\left\| u\right\| =\rho \Big \};$

• (ii) there are $e\in \overline{\sum \nolimits _{i\ge m}E_i}$ with $\left\| e\right\| =1,$ $R>\rho$ and $\epsilon <d$ s.t. if

  $$\begin{aligned} Q=\Big \{ u\in \sum \limits _{i< m}E_i| \left\| u\right\| \le R\Big \} \oplus \{ te:0<t<R\}, \end{aligned}$$

  then ${\mathcal {J}}\le \epsilon$ on $\partial \varOmega ,$ where $\partial \varOmega$ denotes the boundary of Q in $\sum \nolimits _{i< m}E_i\oplus {{\mathbb {R}}}e.$

  Step 1 We give the proof of claim (i).

  By (8),$\forall \ \varepsilon >0,$ $\exists \ \delta ,$ as $|t|<\delta ,$

  $$\begin{aligned} \frac{g(x,t)}{t}<(\lambda _m-\varepsilon )-\lambda _k, \end{aligned}$$

  and then

  $$\begin{aligned} {G(x,t)}\le \frac{t^2}{2}(\lambda _m-\varepsilon -\lambda _k)+C(\varepsilon )|t|^q, \ \ 2<q<2_s^* \end{aligned}$$

  for all real number t and $x\in \varOmega$.

  For $u\in \sum \nolimits _{i\ge m}E_i,$

  $$\begin{aligned} {\mathcal {J}}(u)&= \frac{1}{2}\left\| u\right\| ^2+\frac{1}{2}\int _{\varOmega }a(x)|u|^2 dx-\frac{1}{2}\lambda _k\int _{\varOmega }|u|^2 dx-\int _{\varOmega }G(x,u) dx\\&\ge \frac{1}{2}\left\| u\right\| ^2+\frac{1}{2}\int _{\varOmega }a(x) |u|^2 dx-\frac{1}{2}\lambda _k\int _{\varOmega }|u|^2 dx-C(\varepsilon )\int _{\varOmega }|u|^q dx\\&\quad - \frac{1}{2}(\lambda _m-\varepsilon -\lambda _k)\int _{\varOmega } u^2 dx\\&= \frac{1}{2}\left\| u\right\| ^2+\frac{1}{2}\int _{\varOmega }a(x) |u|^2 dx-\frac{1}{2}(\lambda _m-\varepsilon )\int _{\varOmega }|u|^2 dx-C(\varepsilon )\int _{\varOmega }|u|^q dx\\&\ge \kappa (\varepsilon ) \left\| u\right\| ^2- C(\varepsilon )\int _{\varOmega }|u|^q dx\\&\ge \kappa (\varepsilon ) \left\| u\right\| ^2- C(\varepsilon )\Vert u\Vert ^q \end{aligned}$$

  for some $\kappa (\varepsilon )>0$, where the inequality in Lemma 3 is used. Hence, claim (i) holds.

  Step 2 By (9), it is obvious that ${\mathcal {J}}\le 0$ on $\sum \nolimits _{i\le m-1}E_i.$ If we can prove that

  $$\begin{aligned} \lim _{\left\| u\right\| \rightarrow \infty , u\in \sum \limits _{i\le m}E_i }{\mathcal {J}}(u)=-\infty , \end{aligned}$$

  (21)

  then e can be taken as any element in $E_m$ with $\left\| e\right\| = 1,$ R any number sufficiently large and $\epsilon < d$ any number sufficiently small.

Suppose $m=k$. For $u\in \sum \nolimits _{i\le m}E_i,\ u={\bar{u}}+u^0,$ then

$$\begin{aligned} {\mathcal {J}}(u)&= \frac{1}{2}\left\| \bar{u}\right\| ^2+\frac{1}{2}\int _{\varOmega }a(x)|{\bar{u}}|^2 dx-\frac{1}{2}\lambda _k\int _{\varOmega }|{\bar{u}}|^2 dx-\int _{\varOmega }G(x,u) dx\\&\le \frac{1}{2}(\lambda _{k-1}-\lambda _k)\int _{\varOmega }|\bar{u}|^2 dx-\int _{\varOmega }\Big (G(x,u)-G(x,u^0)\Big ) dx-\int _{\varOmega }G(x,u^0) dx\\&\le \frac{1}{2}(\lambda _{k-1}-\lambda _k+2\varepsilon )\int _{\varOmega }|\bar{u}|^2 dx +C(\varepsilon )\int _{\varOmega }| u^0|^{2\alpha } dx-\int _{\varOmega }G(x,u^0) dx+C(\varepsilon ). \end{aligned}$$

Choosing $0< \varepsilon < \frac{\lambda _k -\lambda _{k-1}}{2}$ and using the condition ($\hbox {G}_+$), we obtain (21).

If $m < k$, then $u ={\bar{u}}$. The proof of (21) is much easier. Hence, the theorem is proved by Rabinowitz’s linking theorem, e.g. see Theorem 2,12 in [20]. $\square$

Proof of Theorem 3

Under the conditions of the theorem, we can prove

• (i) there are $p, d > 0$ such that ${\mathcal {J}}< - d$ on $\Big \{ u\in \sum \nolimits _{i\le m}E_i|\left\| u\right\| =\rho \Big \};$

• (ii) ${\mathcal {J}} \ge 0$ on $\sum \nolimits _{i\ge m+1}E_i;$

• (iii) ${\mathcal {J}}\rightarrow +\infty$ as $u\in \sum \nolimits _{i\ge m}E_i$ and $\left\| u\right\| \rightarrow \infty .$

Then, for $I = - {\mathcal {J}},$ we use Theorem 5.29 in [16] and obtain a positive(nonzero) critical value for I. This completes the proof. $\square$

Proof of Theorem 4

Consider the functional on $X_0$

$$\begin{aligned} {\mathcal {J}}(u)=\frac{1}{2}\left\| u\right\| ^2 +\frac{1}{2}\int _{\varOmega } a(x)u^2(x)dx-\frac{1}{2}\int _{\varOmega } \lambda _ku^2(x)dx -\int _{\varOmega }G(x,u)dx. \end{aligned}$$

Clearly, under our assumptions, ${\mathcal {J}}$ is even and $C^1$ on $X_0$, satisfying P-S condition.

We set $X_2= \sum \nolimits _{i=1}^{k-1}E_i$ as the subspace $X_2$ in Proposition 2. Then $X_2^\perp = \overline{\sum \nolimits _{i\ge k}E_i}.$ For every $u\in X_0,$ we write $u={\bar{u}} + u^0 +{\widetilde{u}}.$ Write the decomposition as $X_0=X_2\oplus X_2^\perp =X_2\oplus E_k \oplus \overline{\sum \nolimits _{i\ge {k+1}}E_i}.$

For $u\in X_2^\perp ,\ u= u^0 +{\widetilde{u}},$ by Lemma 3, we obtain that

$$\begin{aligned} {\mathcal {J}}(u) =&\frac{1}{2}\left\| \widetilde{u}\right\| ^2+\frac{1}{2}\int _{\varOmega }a(x)|{\widetilde{u}}(x)|^2 dx-\frac{1}{2}\int _{\varOmega }\lambda _k|{\widetilde{u}}(x)|^2 dx -\int _{\varOmega }G(x,u^0+{\widetilde{u}}) dx\\ \ge&\kappa \left\| {\widetilde{u}}\right\| ^2 -\int _{\varOmega }\Big [G(x,u^0+{\widetilde{u}})-G(x,{u}^0)\Big ] dx-\int _{\varOmega }G(x,{u}^0) dx. \end{aligned}$$

for some $\kappa >0$. By condition ($\hbox {g}_1$) and a similar argument of (19), we get the ${\mathcal {J}}$ is bounded below on $X_2^\perp$.

Set $X_1=\sum \nolimits _{i=1}^{k}E_i\subset L^\infty (\varOmega )$. Since $X_1$ is finite dimensional, there exists $\delta _2>0$ s.t. $|u|_\infty \le r,$ $\forall u\in X_1$ with $\left\| u\right\| \le \delta _2.$ Hence by ($\hbox {g}_2$),

$$\begin{aligned} {\mathcal {J}}(u)\le -\int _{\varOmega }G(x,{u}) dx<0,\ \ \forall u\in X_1\cap S_{\delta _2}. \end{aligned}$$

Noting that $m-j=\text {dim} X_1-\text {codim}X_2^\perp =\text {dim} E_k=m,$ hence ${\mathcal {J}}$ has at least m pairs of critical points corresponding to negative critical values. This completes the proof. $\square$

Proof of Theorem 5

We suppose that all the eigenvalues of (7) are listed as $\lambda _1<\cdots<\lambda _m<0<\lambda _{m+1}<\cdots<\lambda _n<\cdots$ and the corresponding eigenspaces are denoted by $E_k$ ($k=1,2,\ldots$). Set $X_0^-=\sum \limits _{i\le m}E_i,$ $X_0^+=\overline{\sum \limits _{i\ge m+1}E_i}.$ So $X_0=X_0^-\oplus X_0^+$ and correspondingly $u=u^-+u^+$ for every $u\in X_0$.

$$\begin{aligned} {\mathcal {J}}(u)&=\frac{1}{2}\int _{{\mathbb {R}}^{2N}}\Big [{| (u^-(x)+u^+(x))-(u^-(y)+u^+(y))|^2}{K(x-y)}\Big ]dxdy\\&\quad +\int _{\varOmega } [\frac{1}{2}a(x)u^2-G(x,u)] dx\\&=\frac{1}{2}\int _{{\mathbb {R}}^{2N}}{| u^+(x)- u^+(y)|^2}{K(x-y)}dxdy\\&\quad +\frac{1}{2}\int _{{\mathbb {R}}^{2N}}{| u^-(x)- u^-(y)|^2}{K(x-y)}dxdy\\&\quad +\frac{1}{2}\int _{\varOmega } a(x)u^2dx-\int _{\varOmega }G(x,{u}) dx. \end{aligned}$$

By ($\hbox {g}_4$) and ($\hbox {g}_6$), we have

$$\begin{aligned} |g(x,u)|\le C |u|^{\frac{\sigma +1}{\sigma -1}},\ |u|\ge r_0,\ \text {a.e.} \ x\in \varOmega . \end{aligned}$$

Since $\sigma >\frac{N}{2s},$ we have $\frac{\sigma +1}{\sigma -1}<2_s^*-1.$ Hence ${\mathcal {J}}$ is differentiable on $X_0$. Combining ($\hbox {g}_4$) and the above inequality, we have

$$\begin{aligned} |g(x,u)|\le \varepsilon |u|+C_\varepsilon |u|^{p-1},\ \frac{2\sigma }{\sigma -1}\le p<2_s^*. \end{aligned}$$

(22)

Step 1 We have the following result:

• (i) $\exists \ r>0,$ s.t. $m=\inf {\mathcal {J}}(S_r^+)>0,$ where $S_r^+=\partial B_r \cap X_0^+.$

• (ii) $\exists \ {\widetilde{r}}>0,$ s.t. ${\mathcal {J}}(u)\le 0, \forall u \in \partial Q,$ where $Q=\{u=u^-+ se_{m+1}: u^-\in X_0^-,\ s\ge 0,\ \left\| u\right\| \le {\widetilde{r}} \}$, where $e_{m+1}$ is a nontrivial eigenfunction corresponding to $\lambda _{m+1}$.

$\square$

Proof

The proof of (i) is standard by equality (22) and the embedding theorem ([3]).

$\forall u=u^-+ se_{m+1}$, we have

$$\begin{aligned} {\mathcal {J}}(u)&=\frac{1}{2}\int _{{\mathbb {R}}^{2N}}{| se_{m+1}(x)- se_{m+1}(y)|^2}K(x-y)dxdy\\&\quad +\frac{1}{2}\int _{{\mathbb {R}}^{2N}}{| u^-(x)- u^-(y)|^2}K(x-y)dxdy\\&\quad +\frac{1}{2}\int _{\varOmega } a(x)(u^-)^2dx+\frac{1}{2}\int _{\varOmega } a(x)(se_{m+1})^2dx-\int _{\varOmega }G(x,{u^-+se_{m+1}}) dx. \end{aligned}$$

In order to prove (ii), we follow some arguments in [19] (p.72) and only give the outlines. We just need to prove

$$\begin{aligned} \displaystyle \lim _{u\in \mathrm {span} \{X^-, e_{m+1}\}, \Vert u\Vert \rightarrow \infty }J(u)=-\infty . \end{aligned}$$

Otherwise, there is $M\in {\mathbb {R}}$ and a sequence $u_n\in \mathrm {span} \{X^-, e_{m+1}\}, \Vert u_n\Vert \rightarrow \infty$ such that ${\mathcal {J}}(u_n)\ge M$. We write $u_n= u_n^-+ u_n^+$ and define $w_n=u_n/\Vert u_n\Vert$ with the property $\Vert w_n\Vert =1$ . Noticing that the sequence lies in a finite dimensional space, without loss of generality,we assume that $u_n^-\rightarrow w^-, u_n^+\rightarrow w^+.$ By using the inequality $\frac{M}{\Vert u_n\Vert ^2}\le \frac{{\mathcal {J}}(u_n)}{\Vert u_n\Vert ^2}$ and displaying the right term, with the help of the inequality such as (11), Lemma 3.5 and nonnegativity of G(x, u), we can get that $w^+\ne 0.$ Then the similar proof follows by the obvious existence of ${\bar{\mu }}$ and the choices of $\omega >{\bar{\mu }}$ and $\varOmega$ being the one in our paper.

Noticing also that ${\mathcal {J}}|_{X_0^-} \le 0$, we have that (ii) holds when ${\widetilde{r}}>0$ is sufficiently large. This completes the proof of Step 1.

Step 2 ${\mathcal {J}}$ satisfies the Cerami condition at any level $c\in R.$ That is, if $\{u_n\}$ is any sequence in $X_0$ such that

$$\begin{aligned} {\mathcal {J}}(u_n)\rightarrow c \end{aligned}$$

and

$$\begin{aligned} (1+ \left\| u_n\right\| )\sup \{|<{\mathcal {J}}'(u_n),\phi >:\phi \in X_0,|\left\| \phi \right\| =1\}\rightarrow 0, \end{aligned}$$

then $\{u_n\}$ has a convergent subsequence in $X_0$ [22]. $\square$

Proof

It is clear that for n big enough, we have

$$\begin{aligned} C_0\ge {\mathcal {J}}(u_n)-\frac{1}{2}{\mathcal {J}}'(u_n)u_n=\int _{\varOmega } {\widetilde{G}}(x,u_n)dx. \end{aligned}$$

(23)

We want to get that $\{u_n\}$ is bounded in $X_0.$ Suppose, by contradiction, there is a subsequence of $u_n,$ still denoted by $u_n,$ such that $\left\| u_n\right\| \rightarrow +\infty$ as $n\rightarrow +\infty .$

Set

$$\begin{aligned} v_n=\frac{u_n}{ \left\| u_n\right\| }. \end{aligned}$$

Hence

$$\begin{aligned}&{\mathcal {J}}'(u_n)(u_n^+-u_n^-)\\&\quad =\int _{{\mathbb {R}}^{2N}}{(u_n(x)- u_n(y)) [(u_n^+(x)-u_n^-(x))-( u_n^+(y)-u_n^-(y))]}K(x-y)dxdy\\&\qquad +\int _{\varOmega } a(x)u_n(u_n^+-u_n^-)dx-\int _{\varOmega }g(x,{u_n})(u_n^+-u_n^-) dx\\&\quad =\int _{{\mathbb {R}}^{2N}}|u_n^+(x)- u_n^+(y)|^2 K(x-y)dxdy+\int _{\varOmega } a(x)(u_n^+)^2dx\\&\qquad -\int _{{\mathbb {R}}^{2N}}|u_n^-(x)- u_n^-(y)|^2 K(x-y)dxdy\\&\qquad -\int _{\varOmega } a(x)(u_n^-)^2dx-\int _{\varOmega }g(x,{u_n})(u_n^+-u_n^-) dx\\&\quad \ge \delta \left\| u_n^+\right\| ^2+\delta \left\| u_n^-\right\| ^2\\&\qquad -\int _{\varOmega }g(x,{u_n})(u_n^+-u_n^-) dx\\&\quad \ge \left\| u_n\right\| ^2\Big (\delta -\int _{\varOmega }\frac{g(x,{u_n})(v_n^+-v_n^-)}{\left\| u_n\right\| } dx\Big ), \end{aligned}$$

where $\delta >0$ comes from Lemma 3 and a similar inequality as (11),by noting the finite dimensionality of $X_0^-$. So,

$$\begin{aligned} \liminf _{n\rightarrow \infty }\int _{\varOmega }\frac{g(x,{u_n})(v_n^+-v_n^-)}{\left\| u_n\right\| } dx \ge \delta . \end{aligned}$$

(24)

Set $g(r)=\inf \{{\widetilde{G}}(x,u):x\in {\mathbb {R}}^N,\ |u|\ge r \}.$ Then $g(r)\rightarrow +\infty ,\ \text {as}\ r\rightarrow +\infty$; for the proof see p.45 in [19]. Moreover, by ($\hbox {g}_6$), $g(r)>0,\ \forall r>0$. By (23), we get

$$\begin{aligned} C_0\ge \int _{\varOmega } {\widetilde{G}}(x,u_n)dx\ge \int _{\{x:|u_n(x)|\ge r\}} {\widetilde{G}}(x,u_n)dx\ge g(r)\cdot \text {meas}\{ x:|u_n(x)|\ge r\}. \end{aligned}$$

Hence

$$\begin{aligned} \text {meas}\{ x:|u_n(x)|\ge r\}\le \frac{C_0}{g(r)},\ \forall r>0. \end{aligned}$$

For every $\varepsilon >0,$ choose $a_\varepsilon >r_0$ s.t.

$$\begin{aligned}&\text {meas}\{ x:|u_n(x)|\ge a_\varepsilon \}<\varepsilon . \\&\Big |\int _{\{x\in \varOmega :|u_n(x)|\ge a_\varepsilon \}}\frac{g(x,{u_n})(v_n^+-v_n^-)}{\left\| u_n\right\| } dx\Big |\\&\quad \le C_0\int _{\{x\in \varOmega :|u_n(x)|\ge a_\varepsilon \}}\frac{\widetilde{G}(x,u_n)^{\frac{1}{\sigma }}|u_n||v_n^+-v_n^-|}{\left\| u_n\right\| } dx \ (\text {by} \ \text {g}_6)\\&\quad \le C_0\int _{\{x\in \varOmega :|u_n(x)|\ge a_\varepsilon \}} {\widetilde{G}(x,u_n)^{\frac{1}{\sigma }}|v_n||v_n^+-v_n^-|} dx\\&\quad \le C_0\int _{\{x\in \varOmega :|u_n(x)|\ge a_\varepsilon \}} {\widetilde{G}(x,u_n)^{\frac{1}{\sigma }}|v_n^+|^2} dx+C_0\int _{\{x:|u_n(x)|\ge a_\varepsilon \}} {{\widetilde{G}}(x,u_n)^{\frac{1}{\sigma }}|v_n^-|^2} dx\\&\quad \le C_0\int _{\{x\in \varOmega :|u_n(x)|\ge a_\varepsilon \}} {\widetilde{G}(x,u_n)dx(\Vert v_n^+\Vert _{2^*_s}^{2}+\Vert v_n^-\Vert _{2_s^*}^2}) \cdot \text {m}\{ x\in \varOmega :|u_n(x)|\ge a_\varepsilon \}^{\frac{1}{r}}\le C\varepsilon ^{\frac{1}{r}}, \end{aligned}$$

where the last inequality comes from Höder inequality by choosing $p=\sigma , q=\frac{N}{N-2s}, \frac{1}{r}=1-\frac{1}{p}-\frac{1}{q}$ and C is a constant independent of $\varepsilon$ and n.

From condition ($\hbox {g}_4$), for the above $\varepsilon >0$, we can choose $0<\delta =\delta (\varepsilon )<a_\varepsilon$ s.t.

$$\begin{aligned} |g(x,u)|\le \varepsilon |u| \ \text {for} \ |u|\le \delta . \end{aligned}$$

So, we derive

$$\begin{aligned}&\Big |\int _{\{x\in \varOmega :0\le |u_n(x)|\le \delta \}}\frac{g(x,{u_n})(v_n^+-v_n^-)}{\left\| u_n\right\| } dx\Big |\\&\quad \le \int _{\{x\in \varOmega :0\le |u_n(x)|\le \delta \}}\frac{\varepsilon |u_n||v_n^+-v_n^-|}{\left\| u_n\right\| } dx\\&\quad \le \varepsilon \int _{\varOmega }[(v_n^+)^2+(v_n^-)^2]dx =\varepsilon [|v_n^+|_2^2+|v_n^-|_2^2]\le C\varepsilon , \end{aligned}$$

where C is a constant independent of $\varepsilon$ and n.

We turn to estimate of final part of the integration.

It is clear that $\exists \gamma =\gamma (\varepsilon )$ s.t. $|g(x,u_n)|\le \gamma ,\ \forall x\in {\{x\in \varOmega :\delta<|u_n(x)|< a_\varepsilon \}}.$ Hence, for n large enough, we have

$$\begin{aligned} \Big |\int _{\{x\in \varOmega :\delta<|u_n(x)|< a_\varepsilon \}}\frac{g(x,{u_n})(v_n^+-v_n^-)}{\left\| u_n\right\| } dx\Big | \le \frac{\gamma }{\left\| u_n\right\| }\int _{\varOmega }(|v_n^+|+|v_n^-|)dx \le \frac{C_\varepsilon }{\left\| u_n\right\| }<\varepsilon . \end{aligned}$$

Combining the above arguments, for every $\varepsilon >0$ and we can choose N large enough such that for as $n>N$ we have

$$\begin{aligned} \Big |\int _{\varOmega }\frac{g(x,{u_n})(v_n^+-v_n^-)}{\left\| u_n\right\| } dx\Big |<C\varepsilon +C\varepsilon ^{\frac{1}{r}}, \end{aligned}$$

where C is a constant independent of $\varepsilon$ and n,  which contradicts (24). So $u_n$ is bounded in $X_0.$ A simple argument implies that $\{u_n\}$ contains a convergent subsequence in $X_0.$

Hence, we have obtained both the compactness properties and the geometrical structure of the functional. Hence, by linking theorem, we complete the proof of Theorem 1.5. $\square$