Infinitely Many Solutions for a Fourth-Order Semilinear Elliptic Equations Perturbed from Symmetry

Abstract

In this paper, we study the existence of multiple solutions for the following biharmonic problem

$$\begin{aligned} \Delta ^2 u= & {} f(x,u) + g(x,u)\quad \hbox {in}\quad \Omega ,\\ u= & {} \Delta u =0 \quad \hbox {on}\quad \partial \Omega , \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N, (N > 4)$ is a smooth bounded domain and $f(x,\xi )$ is odd in $\xi , g(x,\xi )$ is a perturbation term. By using the variant of Rabinowitz’s perturbation method, under some growth conditions on f and g, we show that there are infinitely many weak solutions to the problem.

1 Introduction

In the last decades, the biharmonic elliptic equation

$$\begin{aligned} \begin{array}{ll} \Delta ^2 u= f(x,u)&{}\quad \hbox {in}\quad \Omega , \\ u=\Delta u =0 &{}\quad \text {on}\quad \partial \Omega , \end{array} \end{aligned}$$

(1.1)

has been studied by many authors see [4, 6,7,8, 16,17,18] and the references therein.

In this paper, we study the existence of multiple weak solutions to the following problem

$$\begin{aligned} \begin{array}{ll} \Delta ^2 u= f(x,u) + g(x,u)&{}\quad \hbox {in}\quad \Omega , \\ u= \Delta u =0 &{}\quad \hbox {on}\quad \partial \Omega , \end{array} \end{aligned}$$

(1.2)

where $\Omega \subset \mathbb {R}^N, (N > 4)$ is a smooth bounded domain. To study the problem (1.2), we make the following assumptions:

We assume that $f: \Omega \times \mathbb {R} \rightarrow \mathbb {R}$ is a function such that

• (A1) $f(x,\xi )=f_1(x,\xi )+f_2(x,\xi ), f_1, f_2 \in C(\overline{\Omega }\times \mathbb {R}, \mathbb {R})$ and there exist constants $C_1 >0$ and $1< p<2$ such that

  $$\begin{aligned} \left| f_1(x,\xi )\right| \le C_1\left| \xi \right| ^{p-1},\quad (x,\xi ) \in \overline{\Omega }\times \mathbb {R}; \end{aligned}$$

  (1.3)

• (A2) there exist constants $C_2 >0$ and $1<\mu <2$ such that

  $$\begin{aligned} f_1(x,\xi )\xi -\mu F_1(x,\xi )\le 0,\quad (x,\xi )\in \Omega \times \mathbb {R}, \end{aligned}$$

  where $F_1(x,\xi ):=\int \limits _0^{\xi } f_1(x,\tau )\mathrm {d}\tau ;$

• (A3) there exist constants $C_2 >0, 1<p_1<2$ and $2<p_2<2_*$ such that

  $$\begin{aligned} F_1(x,\xi )\ge C_2(\left| \xi \right| ^{p_1} -\left| \xi \right| ^{p_2}),\quad (x,\xi )\in \Omega _0\times \mathbb {R}, \end{aligned}$$

  where $2_*:=\frac{2 N}{ N-4}, \Omega _0$ is a nonempty open and $\Omega _0 \subset \Omega$;

• (A4) there exist constants $C_3 >0$ and $2<p_3<2_*$ such that

  $$\begin{aligned} \left| f_2(x,\xi )\right| \le C_3\left| \xi \right| ^{p_3-1},\quad (x,\xi )\in \overline{\Omega }\times \mathbb {R}; \end{aligned}$$

• (A5) $f_i(x,\xi ) =-f_i(x,-\xi ), i=1,2, (x,\xi ) \in \Omega \times \mathbb {R}$.

Let $g: \Omega \times \mathbb {R} \rightarrow \mathbb {R}$ is a function such that

• (B) $g \in C(\overline{\Omega }\times \mathbb {R}, \mathbb {R})$ and there exist constants $C_4 >0$ and $2<\theta <2_*$ such that

  $$\begin{aligned} \left| g(x,\xi )\right| \le C_4\left| \xi \right| ^{\theta -1},\quad (x,\xi )\in \overline{\Omega }\times \mathbb {R}. \end{aligned}$$

Now, we formulate the main result of this paper.

Theorem 1.1

Assume that (A1)–(A5), (B) are satisfied and

$$\begin{aligned} \frac{2p}{2-p} > \frac{N}{\theta -2}. \end{aligned}$$

(1.4)

Then the problem (1.2) has a sequence of small negative energy solutions converging to zero.

Example 1.2

Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb {R}^5$ and

$$\begin{aligned} f(x,\xi ) =a(x)\left| \xi \right| ^{-\frac{1}{2}} \xi \cos \left| \xi \right| ^{\frac{3}{2}}, \quad g(x,\xi )=\xi ^{\theta -1}, \end{aligned}$$

where $a(x)\in C(\overline{\Omega }, \mathbb {R})$ changes sign in $\Omega , \theta \in (\frac{7}{6}, 10)$. Set

$$\begin{aligned} f_1(x,\xi )=a(x)\left| \xi \right| ^{-\frac{1}{2}}\xi , \quad f_2(x,\xi )=a(x)\left| \xi \right| ^{-\frac{1}{2}}\xi \left( \cos \left| \xi \right| ^{\frac{3}{2}}-1\right) . \end{aligned}$$

Thus, all conditions of Theorem 1.1 are satisfied with

$$\begin{aligned} 2_* =10;\quad p=\mu =p_1=\frac{3}{2};\quad p_2=p_3=3. \end{aligned}$$

By Theorem 1.1, the problem (1.2) has a sequence of small negative energy solutions converging to zero.

2 Proof of Theorem 1.1

Define the Euler–Lagrange functional associated with the problem (1.2) (see, e.g., [13, 14]) as follows

$$\begin{aligned} I\left( u\right) = \frac{1}{2}\int \limits _{\Omega }\left| \Delta u \right| ^2 \mathrm {d}x - \int \limits _\Omega F\left( x,u\right) \mathrm {d}x - \int \limits _\Omega G\left( x,u\right) \mathrm {d}x, \end{aligned}$$

where $F(x,u): = \int _{0}^u f(x,\xi ) \mathrm {d}\xi$ and $G(x,u): = \int _{0}^u g(x,\xi ) \mathrm {d}\xi .$

From (A1), (A4) and (B), we have I is well defined on $H_{0}^{2}( \Omega )$ and $I \in C^1(H_{0}^{2}( \Omega ), \mathbb {R})$ with

$$\begin{aligned} \langle I'(u), v \rangle = \int \limits _{\Omega }\Delta u \Delta v \mathrm {d}x - \int \limits _\Omega f\left( x,u\right) v \mathrm {d}x - \int \limits _\Omega g\left( x,u\right) v \mathrm {d}x \end{aligned}$$

(2.1)

for all $v \in H_{0}^{2}( \Omega ).$ One can also check that the critical points of I are weak solutions of the problem (1.2).

Next, we introduce a cut-off function $\zeta \in C^\infty (\mathbb {R}, \mathbb {R})$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \zeta (\xi )=1, &{} \xi \in (-\infty , 1],\\ 0\le \zeta (\xi )\le 1, &{} \xi \in (1,2),\\ \zeta (\xi ) =0, &{} \xi \in [2,+\infty ), \\ \left| \zeta '(\xi )\right| \le 2&{} \xi \in \mathbb {R}. \end{array}\right. } \end{aligned}$$

(2.2)

With the help of this cut-off function $\zeta$, define

$$\begin{aligned} \pi (u) :=\zeta \left( \frac{\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) ,\quad \forall u\in H_{0}^{2}( \Omega ), \end{aligned}$$

(2.3)

where $T_0$ is a small positive constant independent of u given by (2.10) and (2.49).

Lemma 2.1

The functional $\pi$ defined by (2.3) is of class $C^1(H_{0}^{2}( \Omega ), \mathbb {R})$ and

$$\begin{aligned} \left| \langle \pi '(u), u \rangle \right| \le 8, \quad \forall u \in H_{0}^{2}( \Omega ). \end{aligned}$$

Proof

By direct computation, we get

$$\begin{aligned} \langle \pi '(u), v \rangle =2\zeta '\left( \frac{\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) \frac{(u,v)_{H_{0}^{2}( \Omega )}}{T_0}, \quad \forall u, v \in H_{0}^{2}( \Omega ). \end{aligned}$$

(2.4)

Assume that $u_n\rightarrow u_0$ in $H_{0}^{2}( \Omega ).$ By the definition of $\zeta$ and (2.4), for any $v \in H_{0}^{2}( \Omega ),$ we have that

$$\begin{aligned}&\left| \langle \pi '(u_n) -\pi '(u_0), v \rangle \right| \\&\quad =2 \left| \zeta '\left( \frac{\left\| u_n\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) \frac{(u_n,v)_{H_{0}^{2}( \Omega )}}{T_0} - \zeta ' \left( \frac{\left\| u_0\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) \frac{(u_0,v)_{H_{0}^{2}( \Omega )}}{T_0} \right| \\&\quad \le 2T_0^{-1}\left\| v\right\| _{H_{0}^{2}( \Omega )} \Bigg [\left| \zeta '\left( \frac{\left\| u_n\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) \right| \left\| u_n -u_0\right\| _{H_{0}^{2}( \Omega )} \\&\qquad + \left| \zeta '\left( \frac{\left\| u_n\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) - \zeta '\left( \frac{\left\| u_0\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) \right| \left\| u_0\right\| _{H_{0}^{2}( \Omega )} \Bigg ], \end{aligned}$$

which implies that $\left\| \pi '(u_n) -\pi '(u_0)\right\| _{(H_{0}^{2}( \Omega ))^*}\rightarrow 0, n \rightarrow \infty$. So $\pi \in C^1(H_{0}^{2}( \Omega ), \mathbb {R})$. By (2.2) and (2.4), we get

$$\begin{aligned} \left| \langle \pi '(u), u \rangle \right| \le 8, \quad \forall u \in H_{0}^{2}( \Omega ). \end{aligned}$$

$\square$

With the help of this functional $\pi$, we define a new functional $\overline{I}$ on $H_{0}^{2}( \Omega )$ by

$$\begin{aligned} \overline{I}(u)&= \frac{1}{2}\int \limits _{\Omega }\left| \Delta u \right| ^2 \mathrm {d}x - \int \limits _\Omega F_1\left( x,u\right) \mathrm {d}x - \pi (u)\nonumber \\&\quad \times \left( \int \limits _\Omega F_2\left( x,u\right) \mathrm {d}x + \int \limits _\Omega G\left( x,u\right) \mathrm {d}x\right) , \end{aligned}$$

(2.5)

where $F_2(x,u)=\int _0^u f_2(x,\tau )\mathrm {d}\tau$. From Lemma 2.1, hence $\overline{I}\in C^1(H_{0}^{2}( \Omega ), \mathbb {R})$ and for any $u, v \in H_{0}^{2}( \Omega )$, we have

$$\begin{aligned} \langle \overline{I}'(u), v \rangle&= \int \limits _{\Omega }\Delta u \Delta v \mathrm {d}x -\pi (u)\left( \int \limits _\Omega f_2\left( x,u\right) v \mathrm {d}x + \int \limits _\Omega g\left( x,u\right) v \mathrm {d}x \right) \nonumber \\&\quad - \int \limits _\Omega f_1\left( x,u\right) v \mathrm {d}x- \langle \pi '(u), v \rangle \left( \int \limits _\Omega F_2\left( x,u\right) \mathrm {d}x + \int \limits _\Omega G\left( x,u\right) \mathrm {d}x\right) . \end{aligned}$$

(2.6)

Lemma 2.2

Assume that (A1), (A2), (A4), (B) are satisfied and u is a critical point of $\overline{I}$. Then

$$\begin{aligned} \overline{I}(u) \le \frac{\mu -2}{4\mu } \left\| u\right\| ^2_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.7)

Proof

Consider two cases.

Case 1. Let u is a critical point of $\overline{I}$ and $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}>2T_0$, by the definition of $\zeta$, we have $\pi (u)=0$ and $\pi '(u)=0$. From (A1) and (2.6), we get that

$$\begin{aligned} \overline{I}(u)&= \overline{I}(u) -\mu ^{-1}\langle \overline{I}'(u), u \rangle \nonumber \\&=\frac{\mu -2}{2\mu } \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} +\mu ^{-1}\int \limits _{\Omega }\left( f_1(x,u)u -\mu F_1(x,u)\right) \mathrm {d}x \nonumber \\&\le \frac{\mu -2}{4\mu } \left\| u\right\| ^2_{H_{0}^{2}( \Omega )},\quad \hbox {by}\quad 1<\mu <2\quad \hbox {and}\quad f_1(x,u)u -\mu F_1(x,u)\le 0. \end{aligned}$$

(2.8)

Case 2. Let u is a critical point of $\overline{I}$ and $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}\le 2T_0$. By applying embedding inequalities, Lemma 2.1, (A2), (A4) and (B), we get that

$$\begin{aligned} \overline{I}(u)&= \overline{I}(u) -\mu ^{-1}\langle \overline{I}'(u), u \rangle \nonumber \\&\le \frac{\mu -2}{2\mu } \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} + \frac{\mu +9}{\mu }C_3\left\| u\right\| ^{p_3}_{L^{p_3}(\Omega )} + \frac{\mu +9}{\mu }C_4\left\| u\right\| ^{\theta }_{L^{\theta }(\Omega )} \nonumber \\&\le \frac{\mu -2}{2\mu } \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} + \frac{\mu +9}{\mu }C_3 C_{p_3}^{p_3}\left\| u\right\| ^{p_3}_{H_{0}^{2}( \Omega )}\nonumber \\&\quad + \frac{\mu +9}{\mu }C_4C_{\theta }^{\theta }\left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}, \end{aligned}$$

(2.9)

where $C_{p_3}, C_\theta$ are constants such that

$$\begin{aligned} \left\| u\right\| _{L^{p_3}(\Omega )} \le C_{p_3} \left\| u\right\| _{H_{0}^{2}( \Omega )}, \left\| u\right\| _{L^{\theta }(\Omega )} \le C_\theta \left\| u\right\| _{H_{0}^{2}( \Omega )}. \end{aligned}$$

Since $p_3>2, p_4>2$, we can choose $T_0$ small enough such that if $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}\le 2T_0$

$$\begin{aligned} \frac{\mu +9}{\mu }C_3 C_{p_3}^{p_3}\left\| u\right\| ^{p_3}_{H_{0}^{2}( \Omega )} + \frac{\mu +89}{\mu }C_4 C_{\theta }^{\theta }\left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )} \le \frac{2-\mu }{4\mu } \left\| u\right\| ^2_{H_{0}^{2}( \Omega )}, \end{aligned}$$

(2.10)

By (2.8) and (2.10), we get the conclusion of the lemma. $\square$

Next, we introduce a cut-off function $\chi \in C^\infty (\mathbb {R}, \mathbb {R})$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \chi (\xi )=1, &{} \xi \in (-\infty , \frac{A}{2}],\\ 0\le \chi (\xi )\le 1, &{} \xi \in (\frac{A}{2},\frac{A}{4}),\\ \chi (\xi ) =0, &{} \xi \in [\frac{A}{4},+\infty ), \\ \left| \chi '(\xi )\right| \le -8A^{-1}, &{} \xi \in \mathbb {R}, \quad A:=\frac{\mu -2}{4\mu }. \end{array}\right. } \end{aligned}$$

We put

$$\begin{aligned}&\ell (u):=\chi \left( \left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u)\right) ,\quad \forall u \in H_{0}^{2}( \Omega ) \backslash \{0\},\end{aligned}$$

(2.11)

$$\begin{aligned}&\psi (u):= {\left\{ \begin{array}{ll} \pi (u)\ell (u)\int _{\Omega }G(x,u)\mathrm {d}x,&{} \quad \forall u \in H_{0}^{2}( \Omega ) \backslash \{0\},\\ 0,&{}\quad u=0, \end{array}\right. } \end{aligned}$$

(2.12)

and

$$\begin{aligned} J(u)&= \frac{1}{2}\int \limits _{\Omega }\left| \Delta u \right| ^2 \mathrm {d}x - \int \limits _\Omega F_1\left( x,u\right) \mathrm {d}x - \pi (u)\int \limits _\Omega F_2\left( x,u\right) \mathrm {d}x \nonumber \\&\quad - \psi (u),\forall u \in H_{0}^{2}( \Omega ). \end{aligned}$$

(2.13)

From (A1), (A4) and (B), it is easy to verify that $\ell (u)\in C^1(H_{0}^{2}( \Omega ) \backslash \{0\}, \mathbb {R})$. By direct computation, for $u \in H_{0}^{2}( \Omega ) \backslash \{0\}$ and for any $v \in H_{0}^{2}( \Omega ),$ we have

$$\begin{aligned} \langle \ell '(u), v \rangle&=\chi '\left( \left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u)\right) \left\| u\right\| ^{-4}_{H_{0}^{2}( \Omega )} \nonumber \\&\quad \times \left( \left\| u\right\| ^{2}_{H_{0}^{2}( \Omega )} \langle \overline{I}'(u), v \rangle -2 \overline{I}(u)(u,v)_{H_{0}^{2}( \Omega )}\right) . \end{aligned}$$

(2.14)

Lemma 2.3

Assume that (A1), (A2), (A4), (B) are satisfied. Then the functional $\psi$ defined by (2.12) is of class $C^1(H_{0}^{2}( \Omega ), \mathbb {R})$ and

$$\begin{aligned} \left| \langle \psi '(u),u\rangle \right| \le 89 C_4C_{\theta }^{\theta }\left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}, \forall u \in H_{0}^{2}( \Omega ). \end{aligned}$$

(2.15)

Proof

For $u=0$ for any $v\in H_{0}^{2}( \Omega )$, by (2.3), (2.12) and (B), we get

$$\begin{aligned} \left| \langle \psi '(0),v\rangle \right| =\lim \limits _{t\rightarrow 0} \left| \frac{\psi (t v) -\psi (0)}{t}\right| \le C_4 \lim \limits _{t\rightarrow 0} \left| t\right| ^{\theta -1}\int \limits _{\Omega }\left| v(x)\right| ^\theta \mathrm {d}x =0; \end{aligned}$$

hence, $\psi '(0)=0$. From (2.4), (2.12), (2.14) and (B) for $u \in H_{0}^{2}( \Omega ) \backslash \{0\}$ and $v\in H_{0}^{2}( \Omega )$, we have that

$$\begin{aligned} \langle \psi '(u),v\rangle&=\langle \pi '(u),v\rangle \ell (u)\int \limits _{\Omega }G(x,u)\mathrm {d}x + \pi (u)\langle \ell '(u),v\rangle \int \limits _{\Omega }G(x,u)\mathrm {d}x \nonumber \\&\quad + \pi (u)\ell (u)\int \limits _{\Omega }g(x,u)v\mathrm {d}x. \end{aligned}$$

(2.16)

Next, we prove $\psi \in C^1(H_{0}^{2}( \Omega ), \mathbb {R})$. Suppose that $u_n \rightarrow u_0$ in $H_{0}^{2}( \Omega ).$ We consider two possible cases.

Case 1. $u_0\ne 0$. By Lemma 2.1, (2.14) and (B), we have

$$\begin{aligned} \psi '(u_n)\rightarrow \psi '(u_0)\quad \hbox {as}\quad n \rightarrow \infty . \end{aligned}$$

Case 2. $u_0=0$. Without loss of generality, we can assume $\left\| u_n\right\| ^2_{H_{0}^{2}( \Omega )}<T_0$. Hence, by (2.2), (2.3) we get $\pi (u_n)=1$ and $\pi '(u_n) =0$; hence,

$$\begin{aligned} \langle \psi '(u_n),v\rangle =\langle \ell '(u_n),v\rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x + \ell (u_n)\int \limits _{\Omega }g(x,u_n)v\mathrm {d}x. \end{aligned}$$

(2.17)

On the other hand, by (2.14), we obtain

$$\begin{aligned} \langle \ell '(u_n),v\rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x&= \chi '\left( \left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u_n)\right) \left\| u_n\right\| ^{-2}_{H_{0}^{2}( \Omega )} \langle \overline{I}'(u_n), v \rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x \\&\quad - 2 \chi '\left( \left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u_n)\right) \left\| u_n\right\| ^{-4}_{H_{0}^{2}( \Omega )}\overline{I}(u_n)(u_n,v)_{H_{0}^{2}( \Omega )}\\&\quad \int \limits _{\Omega }G(x,u_n)\mathrm {d}x. \end{aligned}$$

From the definition of $\chi$ and (B), applying embedding inequalities, we get that

$$\begin{aligned}&\left| \chi '\left( \left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u_n)\right) \left\| u_n\right\| ^{-2}_{H_{0}^{2}( \Omega )} \langle \overline{I}'(u_n), v \rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x\right| \nonumber \\&\quad \le C_5 \left\| \overline{I}'(u_n)\right\| _{(H_{0}^{2}( \Omega ))^*} \left\| v\right\| _{H_{0}^{2}( \Omega )} \left\| u_n\right\| ^{\theta -2}_{H_{0}^{2}( \Omega )},\end{aligned}$$

(2.18)

$$\begin{aligned}&\left| 2 \chi '\left( \left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u_n)\right) \left\| u_n\right\| ^{-4}_{H_{0}^{2}( \Omega )}\overline{I}(u_n)(u_n,v)_{H_{0}^{2}( \Omega )} \int \limits _{\Omega }G(x,u_n)\mathrm {d}x \right| \nonumber \\&\quad \le C_6 \left\| u_n\right\| ^{\theta -1}_{H_{0}^{2}( \Omega )} \left\| v\right\| _{H_{0}^{2}( \Omega )},\end{aligned}$$

(2.19)

$$\begin{aligned}&\left| \ell (u_n)\int \limits _{\Omega }g(x,u_n)v\mathrm {d}x \right| \le C_7 \left\| u_n\right\| ^{\theta -1}_{H_{0}^{2}( \Omega )} \left\| v\right\| _{H_{0}^{2}( \Omega )}, \end{aligned}$$

(2.20)

where $C_j >0, j=5,6,7$, and $(H_{0}^{2}( \Omega ))^*$ denotes the dual space of $H_{0}^{2}( \Omega )$. Since $\pi (u_n)=1, \pi '(u_n) =0$ and $u_n\rightarrow 0, n\rightarrow \infty$, we have that

$$\begin{aligned} \left\| \overline{I}'(u_n)\right\| _{(H_{0}^{2}( \Omega ))^*} \rightarrow 0,\quad \hbox {as}\quad n\rightarrow \infty . \end{aligned}$$

(2.21)

From (2.17)–(2.21), we see that

$$\begin{aligned}&\left\| \psi '(u_n)-\psi '(0)\right\| _{(H_{0}^{2}( \Omega ))^*}\\&\quad =\sup \limits _{\left\| v\right\| _{H_{0}^{2}( \Omega )}\le 1} \left| \langle \ell '(u_n),v\rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x + \ell (u_n)\int \limits _{\Omega }g(x,u_n)v\mathrm {d}x \right| \rightarrow 0, \quad \hbox {as}\quad n \rightarrow \infty ; \end{aligned}$$

hence, the continuity of $\psi '$ follows. So we have $\psi \in C^1(H_{0}^{2}( \Omega ), \mathbb {R})$.

Next, we prove (2.15).

If $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}> 2T_0$ or $u=0$ then by (2.2), Lemma 2.1 and (2.16), we have $\langle \psi '(u),u\rangle =0$. Otherwise, $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}\le 2T_0$ and $u\ne 0$. Arguing similarly as in (2.9), we obtain

$$\begin{aligned}&\left| \overline{I}(u) -\mu ^{-1}\langle \overline{I}'(u), u \rangle \right| \nonumber \\&\quad \le 2\left| A\right| \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} + \frac{\mu +9}{\mu }C_3 C_{p_3}^{p_3}\left\| u\right\| ^{p_3}_{H_{0}^{2}( \Omega )} + \frac{\mu +9}{\mu }C_4 C_{\theta }^{\theta }\left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.22)

From (2.10) and (2.22), we have that

$$\begin{aligned} \left| \langle \overline{I}'(u), u \rangle \right| \le \mu \left( 3\left| A\right| \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} + \left| \overline{I}(u)\right| \right) . \end{aligned}$$

(2.23)

By the definition of $\chi$, we have that if $\left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u) \notin [ \frac{A}{2},\frac{A}{4}]$ then $\ell '(u)=0$. Otherwise, if

$$\begin{aligned} \frac{A}{2} \le \left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u) \le \frac{A}{4} \end{aligned}$$

then

$$\begin{aligned} \left| \overline{I}(u) \right| \le \left| A\right| \left\| u\right\| ^{2}_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.24)

In combination with (2.14), (2.23) and (2.24), we get

$$\begin{aligned} \left| \pi (u) \langle \ell '(u), u \rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x\right| \le 80 C_4 C_{\theta }^\theta \left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.25)

By Lemma 2.1 and (2.2), (2.11), we conclude that

$$\begin{aligned}&\left| \langle \pi '(u),u\rangle \ell (u)\int \limits _{\Omega }G(x,u)\mathrm {d}x \right. \nonumber \\&\quad \left. +\,\pi (u)\ell (u)\int \limits _{\Omega }g(x,u)u\mathrm {d}x\right| \le 9 C_4 C_{\theta }^\theta \left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.26)

Combining with (2.16), (2.25) and (2.26), we get (2.15). The proof of Lemma 2.3 is complete. $\square$

Lemma 2.4

Assume that (A1), (A2), (A4), (A5), (B) are satisfied. Then

• (K1) The functional J defined by (2.13) is of class $C^1(H_{0}^{2}( \Omega ), \mathbb {R})$ and there exists a constants $C_{8}$ independent of u such that

  $$\begin{aligned} \left| J(u)-J(-u)\right| \le C_{8}\left| J(u)\right| ^{\frac{\theta }{2}},\quad \forall u \in H_{0}^{2}( \Omega ). \end{aligned}$$

  (2.27)

• (K2) J has no critical point with critical value on $H_{0}^{2}( \Omega )$ and $K_0=\{0\}$, where $K_0:=\{u\in H_{0}^{2}( \Omega ): J(u)=0, J'(u)=0\}.$

Proof

By Lemmas 2.1, 2.3, (A1) and (A4), we have $J \in C^1(H_{0}^{2}( \Omega ), \mathbb {R})$ and

$$\begin{aligned} \langle J'(u),v\rangle&= \int \limits _{\Omega }\Delta u \Delta v \mathrm {d}x - \int \limits _\Omega f_1\left( x,u\right) v \mathrm {d}x -\pi (u)\int \limits _\Omega f_2 \left( x,u\right) v \mathrm {d}x \nonumber \\&\quad - \langle \pi '(u), v \rangle \int \limits _\Omega F_2\left( x,u\right) \mathrm {d}x -\langle \psi '(u),v\rangle , \quad \forall u,v\in H_{0}^{2}( \Omega ). \end{aligned}$$

(2.28)

Next, we prove (2.27). We consider two possible cases.

Case 1. If $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )} > 2T_0$ or $\left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u) > \frac{A}{4}$, by the definition of $\zeta$ and $\chi$ we have $\psi (u)=0$. Then (2.27) holds by (A5) and (2.13).

Case 2. If $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )} \le 2T_0$ and $\left\| u\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u) \le \frac{A}{4}$,

$$\begin{aligned} \left| \overline{I}(u)\right| \ge \frac{\left| A\right| }{4}\left\| u\right\| ^{2}_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.29)

From (B), (2.3), (2.10), (2.11) and (2.29), we get that

$$\begin{aligned} \left| J(u)\right|&\ge \left| \overline{I}(u)\right| - 2\left| \int \limits _\Omega G(x,u)\mathrm {d}x \right| \ge \frac{\left| A\right| }{4}\left\| u\right\| ^{2}_{H_{0}^{2}( \Omega )}-2C_4C_{\theta }^\theta \left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}\\&\ge C_{9} \left\| u\right\| ^{2}_{H_{0}^{2}( \Omega )}; \end{aligned}$$

hence,

$$\begin{aligned} \left| J(u)\right| ^{\frac{\theta }{2}} \ge C_{10}\left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.30)

In view of (A5), (B), (2.3) and (2.11), we obtain

$$\begin{aligned} \left| J(u)-J(-u)\right| \le 2\left| \int \limits _\Omega G(x,u)\mathrm {d}x \right| \le 2C_4C_{\theta }^\theta \left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.31)

It follows from (2.30) and (2.31) that (2.27) holds.

Next, we prove (K2) by contradiction. If $u_0$ is a critical point of J with $J(u_0)>0$, by (A1), (A4) and (B) we get $u_0\ne 0.$ We consider two possible cases.

Case 1. If $\left\| u_0\right\| ^2_{H_{0}^{2}( \Omega )}> 2T_0$ then

$$\begin{aligned} \pi (u_0)=\pi '(u_0)=\psi '(u_0)=0. \end{aligned}$$

By (A2), (2.13) and (2.28), we have that

$$\begin{aligned} 0\le J(u_0)&=J(u_0) -\frac{1}{\mu } \langle J'(u_0),u_0 \rangle \\&=\frac{\mu -2}{2\mu }\int \limits _\Omega \left| \Delta u _0\right| ^2\mathrm {d}x + \frac{1}{\mu }\int \limits _\Omega \left( f_1(x,u_0)u_0-\mu F_1(x,u_0)\right) \mathrm {d}x <0, \end{aligned}$$

which yields a contradiction.

Case 2. If $\left\| u_0\right\| ^2_{H_{0}^{2}( \Omega )} \le 2T_0$ then by Lemmas 2.1, 2.3, (2.10), (2.13) and (2.28), we obtain

$$\begin{aligned} 0\le J(u_0)=J(u_0) -\frac{1}{\mu } \langle J'(u_0),u_0 \rangle \le \frac{\mu -2}{4\mu } \left\| u_0\right\| ^2_{H_{0}^{2}( \Omega )}<0, \end{aligned}$$

which yields a contradiction. Moreover, by a similar proof and direct computation we obtain $K_0=\{0\}$. $\square$

Lemma 2.5

Assume that (A1), (A4), (B) are satisfied. Then the functional J satisfies the Palais–Smale condition.

Proof

Without loss of generality, assume $\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}> 2T_0$. Then, by the definition of $\pi$ and (A1) we obtain

$$\begin{aligned} J(u)\ge \frac{1}{2} \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} -C_{11} \left\| u\right\| ^{p}_{H_{0}^{2}( \Omega )}. \end{aligned}$$

(2.32)

Since $1<p<2$, (2.31) implies that

$$\begin{aligned} J(u) \rightarrow +\infty \hbox { as } \left\| u\right\| _{H_{0}^{2}( \Omega )} \rightarrow +\infty . \end{aligned}$$

(2.33)

Next, we show that J(u) satisfies the Palais–Smale condition. Assume that $\{u_n\}_{n=1}^{\infty } \subset H_{0}^{2}( \Omega )$ is a Palais–Smale sequence, i.e., $\{J(u_n)\}_{n\in \mathbb {N}}$ is bounded and

$$\begin{aligned} J'(u_n) \rightarrow 0 \hbox { as } n \rightarrow +\infty . \end{aligned}$$

Since (2.33), $\{u_n\}_{n=1}^\infty$ is bounded in $H_{0}^{2}( \Omega )$. Therefore, we can (by passing to a subsequence, we can always suppose $u_n\ne 0$ for all n, otherwise, the thesis is obvious) suppose that

$$\begin{aligned} u_n&\rightharpoonup&u_0 \hbox { weakly in } H_{0}^{2}( \Omega ) \hbox { as } n \rightarrow \infty \nonumber \\ u_n\rightarrow & {} u_0 \hbox { a.e., in } \Omega \hbox { as } n \rightarrow \infty \nonumber \\ u_n\rightarrow & {} u_0 \hbox { strongly in } L^q(\Omega ), 1\le q < 2_* \hbox { as } n \rightarrow \infty . \end{aligned}$$

(2.34)

Since (2.34), by (A1), (A4), (B) and standard arguments we get

$$\begin{aligned} \int \limits _\Omega f_i(x,u_n)(u_n -u_0) \mathrm {d}x&\rightarrow 0 \text{ as } n \rightarrow \infty , i=1, 2,\end{aligned}$$

(2.35)

$$\begin{aligned} \int \limits _\Omega g(x,u_n)(u_n -u_0) \mathrm {d}x&\rightarrow 0 \text{ as } n \rightarrow \infty . \end{aligned}$$

(2.36)

From (2.34), we have

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\left| \langle J'(u_n),u_n-u_0\rangle \right| =0. \end{aligned}$$

(2.37)

Next, we distinguish two cases.

Case 1. If $\left\| u_n\right\| ^2_{H_{0}^{2}( \Omega ) } >2T_0$, from (2.2), (2.3), (2.10), (2.12), we get that

$$\begin{aligned} \pi (u_n) =0, \pi '(u_n)=0, \psi '(u_n)=0. \end{aligned}$$

By (2.28), we obtain

$$\begin{aligned} \langle J'(u_n)-J'(u_0),u_n-u_0\rangle&= \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } - \int \limits _\Omega ( f_1\left( x,u_n\right) \nonumber \\&\quad -f_1\left( x,u_0\right) )(u_n-u_0) \mathrm {d}x. \end{aligned}$$

(2.38)

Case 2. If $\left\| u_n\right\| ^2_{H_{0}^{2}( \Omega ) } \le 2T_0$, from (2.28), we have

$$\begin{aligned}&\langle J'(u_n),u_n-u_0\rangle =\left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) }+ \int \limits _{\Omega }\Delta u _0 \Delta (u_n-u_0) \mathrm {d}x\nonumber \\&\quad -\int \limits _\Omega f_1\left( x,u_n\right) (u_n-u_0) \mathrm {d}x \nonumber \\&-\pi (u_n)\int \limits _\Omega f_2\left( x,u_n\right) (u_n-u_0) \mathrm {d}x - \langle \pi '(u_n), u_n-u_0 \rangle \int \limits _\Omega F_2\left( x,u_n\right) \mathrm {d}x\nonumber \\&\quad -\langle \psi '(u_n),u_n-u_0\rangle . \end{aligned}$$

(2.39)

By (A4), (2.2) and (2.4), we get that

$$\begin{aligned}&\left| \langle \pi '(u_n), u_n-u_0 \rangle \int \limits _\Omega F_2\left( x,u_n\right) \mathrm {d}x \right| \nonumber \\&\quad \le C_3C_{p_3}^{p_3} \left\| u_n\right\| ^{p_3}_{H_{0}^{2}( \Omega )} \left| 2\zeta '\left( \frac{\left\| u_n\right\| ^2_{H_{0}^{2}( \Omega )}}{T_0}\right) \frac{(u_n,u_n-u_0)_{H_{0}^{2}( \Omega )}}{T_0}\right| \nonumber \\&\quad \le 2^{\frac{p_3+4}{2}}C_3 C_{p_3}^{p_3} T_0^{\frac{p_3-2}{2}} \left( \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } + (u_0,u_n-u_0)_{H_{0}^{2}( \Omega )}\right) . \end{aligned}$$

(2.40)

On the other hand, from (2.16), we obtain

$$\begin{aligned} \langle \psi '(u_n),u_n-u_0 \rangle&=\langle \pi '(u_n),u_n-u_0\rangle \ell (u_n)\int \limits _{\Omega }G(x,u_n)\mathrm {d}x \nonumber \\&\quad + \pi (u_n)\langle \ell '(u_n),u_n-u_0\rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x\nonumber \\&\quad + \pi (u_n)\ell (u_n)\int \limits _{\Omega }g(x,u_n)(u_n-u_0)\mathrm {d}x. \end{aligned}$$

(2.41)

Moreover, by (B), (2.4) and (2.11), we obtain

$$\begin{aligned}&\left| \langle \pi '(u_n),u_n-u_0\rangle \ell (u_n)\int \limits _{\Omega }G(x,u_n) \mathrm {d}x \right| \nonumber \\&\quad \le 2^{\frac{\theta +4}{2}}C_4 C_{\theta }^{\theta } T_0^{\frac{\theta -2}{2}} \left( \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } + (u_0,u_n-u_0)_{H_{0}^{2}( \Omega )}\right) . \end{aligned}$$

(2.42)

From the defined of $\chi$, (B), (2.4) and (2.14), we have that

$$\begin{aligned}&\pi (u_n)\langle \ell '(u_n),u_n-u_0\rangle \int \limits _{\Omega }G(x,u_n)\mathrm {d}x =\pi (u_n)\chi '\left( \left\| u_n\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u_n)\right) \left\| u_n\right\| ^{-4}_{H_{0}^{2}( \Omega )}\nonumber \\&\quad \left( \left\| u_n\right\| ^{2}_{H_{0}^{2}( \Omega )} \langle \overline{I}'(u_n), u_n-u_0 \rangle -2 \overline{I}(u_n)(u_n,u_n-u_0)_{H_{0}^{2} ( \Omega )}\right) \nonumber \\&\quad \times \int \limits _{\Omega }G(x,u_n)\mathrm {d}x. \end{aligned}$$

(2.43)

From (2.6), we have

$$\begin{aligned} \langle \overline{I}'(u_n), u_n-u_0 \rangle&= \int \limits _{\Omega }\Delta u _n \Delta (u_n-u_0) \mathrm {d}x- \int \limits _\Omega f_1\left( x,u_n\right) (u_n-u_0) \mathrm {d}x\nonumber \\&\quad -\pi (u_n)\left( \int \limits _\Omega f_2\left( x,u_n\right) (u_n-u_0) \mathrm {d}x + \int \limits _\Omega g\left( x,u_n\right) (u_n-u_0) \mathrm {d}x \right) \nonumber \\&\quad - \langle \pi '(u_n), u_n-u_0 \rangle \left( \int \limits _\Omega F_2\left( x,u_n\right) \mathrm {d}x + \int \limits _\Omega G\left( x,u_n\right) \mathrm {d}x\right) . \end{aligned}$$

(2.44)

By (A4), (B), (2.40), (2.42) and (2.44), we have

$$\begin{aligned} \left| \langle \overline{I}'(u_n), u_n-u_0 \rangle \right|&\le \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) }+ \left| \langle \pi '(u_n), u_n-u_0 \rangle \int \limits _\Omega F_2\left( x,u_n\right) \mathrm {d}x \right| \nonumber \\&\quad + \left| \langle \pi '(u_n), u_n-u_0 \rangle \int \limits _\Omega G\left( x,u_n\right) \mathrm {d}x\right| +o_n(1) \nonumber \\&\le \left( 1+C_{12}\right) \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } + o_n(1), \end{aligned}$$

(2.45)

where $o_n(1)\rightarrow 0$ as $n\rightarrow \infty$ and

$$\begin{aligned} C_{12}= 2^{\frac{p_3+4}{2}}C_3 C_{p_3}^{p_3} T_0^{\frac{p_3-2}{2}} + 2^{\frac{\theta +4}{2}}C_4 C_{\theta }^{\theta } T_0^{\frac{\theta -2}{2}}. \end{aligned}$$

Hence,

$$\begin{aligned}&\left| \pi (u_n)\chi '\left( \left\| u_n\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u_n)\right) \left\| u_n\right\| ^{-2}_{H_{0}^{2}( \Omega )} \langle \overline{I}'_{T_0}(u_n), u_n-u_0 \rangle \int \limits _\Omega G\left( x,u_n\right) \mathrm {d}x \right| \nonumber \\&\quad \le 2^{\frac{\theta +4}{2}} \left| A\right| ^{-1}C_4 C^{\theta }_{\theta }T_0^{\frac{\theta -2}{2}}\left( 1+C_{12}\right) \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) }+ o_n(1).\end{aligned}$$

(2.46)

$$\begin{aligned}&\left| \pi (u_n)\chi '\left( \left\| u_n\right\| ^{-2}_{H_{0}^{2}( \Omega )} \overline{I}(u_n)\right) \left\| u_n\right\| ^{-4}_{H_{0}^{2}( \Omega )}2 \overline{I}(u_n)(u_n,u_n-u_0)_{H_{0}^{2} ( \Omega )}\int \limits _\Omega G\left( x,u_n\right) \mathrm {d}x \right| \nonumber \\&\quad \le 2^{\frac{\theta +4}{2}}C_4 C^{\theta }_{\theta }T_0^{\frac{\theta -2}{2}} \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } + o_n(1). \end{aligned}$$

(2.47)

By (2.41), (2.42), (2.46) and (2.47), we have

$$\begin{aligned} \left| \langle \psi '(u_n),u_n-u_0 \rangle \right| \le C_{13} \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } + o_n(1), \end{aligned}$$

(2.48)

where

$$\begin{aligned} C_{13}= 2^{\frac{\theta +4}{2}}C_4 C_{\theta }^{\theta } T_0^{\frac{\theta -2}{2}} + 2^{\frac{\theta +4}{2}} \left| A\right| ^{-1}C_4 C^{\theta }_{\theta }T_0^{\frac{\theta -2}{2}}\left( 1+C_{12}\right) . \end{aligned}$$

Since $p_3>2$ and $\theta >2,$ we can choose $T_0$ small enough such that

$$\begin{aligned} 2^{\frac{p_3+4}{2}}C_3 C_{p_3}^{p_3} T_0^{\frac{p_3-2}{2}} +C_{13} \le \frac{1}{2}. \end{aligned}$$

(2.49)

By (2.39), (2.40), (2.48) and (2.49), we obtain

$$\begin{aligned} \left| \langle J'(u_n),u_n-u_0\rangle \right|&\ge \left( 1 -2^{\frac{p_3+4}{2}}C_3 C_{p_3}^{p_3} T_0^{\frac{p_3-2}{2}} -C_{13}\right) \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } + o_n(1)\nonumber \\&\ge \frac{1}{2} \left\| u_n-u_0\right\| ^2_{H_{0}^{2}( \Omega ) } + o_n(1). \end{aligned}$$

(2.50)

It follows from (2.37) and (2.50) that $u_n\rightarrow u_0$ as $n\rightarrow \infty .$ The proof of Lemma 2.5 is complete. $\square$

Now, we can show that J has a sequence of critical values. For the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u = \lambda u&{}\hbox {in}\quad \Omega , \\ u= \Delta u =0 &{}\hbox {on}\quad \partial \Omega , \end{array}\right. } \end{aligned}$$

(2.51)

we can show that the Dirichlet eigenvalue the problem (2.51) has a sequence of discrete eigenvalues $\{\lambda _k\}_{k=1}^\infty$ which satisfy

$$\begin{aligned} 0< \lambda _1 < \lambda _2 \le \lambda _3 \le \cdots \le \lambda _k \le \cdots \rightarrow \infty \hbox { as } k \rightarrow \infty , \end{aligned}$$

and $e_1, e_2, \dots$ denote the corresponding eigenfunctions normalized such that $\left\| e_j\right\| _{H_{0}^{2}( \Omega ) } =1,$ for all $j = 1, 2, \ldots$. For any $k>0$, we put $\mathbb {V}_{k} =\hbox {span}\{e_j; j \le k\}$ in $H_{0}^{2}( \Omega )$, $\mathbb {V}_{k}^\bot$ be the orthogonal complement of $\mathbb {V}_{k}$ in $H_{0}^{2}( \Omega )$.

Lemma 2.6

There exists a normalized orthogonal sequence $\{\varphi _k\}_{k=1}^\infty \subset C^\infty _0(\Omega )$ such that $\hbox {supp}~\varphi _k \subset \Omega _0, k\in \mathbb {N}$, where $\Omega _0$ is the nonempty open set given in (A3).

Proof

By (A3), there exist $x_0 \in \Omega _0$ and $\delta _0>0$ such that $B(x_0,\delta _0):=\{x\in \mathbb {R}^N: \left| x-x_0\right| < \delta _0\}\subset \Omega _0$. Choose a strictly increasing sequence $\{\rho _k\}_{k=1}^\infty$ such that

$$\begin{aligned} 0< \rho _1<\rho _2< \cdots< \rho _k < \cdots \rightarrow \frac{\delta _0}{4}. \end{aligned}$$

Define

$$\begin{aligned} O_k: =B(x_0,\rho _{k+1})\backslash {\overline{B}(x_0,\rho _k)},\quad k \in \mathbb {N}. \end{aligned}$$

Let $x_k \in O_k$ and choose $r_k >0$ such that

$$\begin{aligned} B(x_0,r_k) \subset O_k, \quad k\in \mathbb {N}. \end{aligned}$$

(2.52)

Set

$$\begin{aligned} \varphi _0(x):= {\left\{ \begin{array}{ll} e^{\frac{1}{\left| x\right| ^2-1}}&{} \hbox {if}\quad \left| x\right| <1, \\ 0&{}\hbox {if}\quad \left| x\right| \ge 1. \end{array}\right. } \end{aligned}$$

(2.53)

By (2.53), define $\varphi _k$ as follows

$$\begin{aligned} \varphi _k(x):=\varphi _0\left( (x-x_k)/r_k\right) , \quad k \in \mathbb {N}. \end{aligned}$$

(2.54)

By (2.53) and (2.54), we get

$$\begin{aligned} \varphi _k \in C^\infty _0(\Omega ), \quad k \in \mathbb {N}. \end{aligned}$$

Moreover, from (2.52)-(2.54), we have

$$\begin{aligned} \hbox {supp}\,\varphi _k \subset O_k\subset \Omega _0, \quad k \in \mathbb {N}. \end{aligned}$$

Then the supports of $\varphi _k$ are disjoint to each other, which implies that $\{\varphi _k\}_{k=1}^\infty$ form a linearly independent sequence in $H_{0}^{2}( \Omega )$. By Gram–Schmidt orthogonalization process, there exists a normalized orthogonal sequence also denoted by $\{\varphi _k\}_{k=1}^\infty$ in $H_{0}^{2}( \Omega )$ and

$$\begin{aligned} \hbox {supp}\,\varphi _k \subset \Omega _0, k\in \mathbb {N}. \end{aligned}$$

$\square$

With the help of the normalized orthogonal sequence $\{\varphi _k\}_{k=1}^\infty$, define some subspaces as follows:

$$\begin{aligned}&\mathbb {W}_k:= \hbox {span}\{\varphi _j; j \le k\},\\&\quad B_k:=\{u\in \mathbb {W}_{k}: \left\| u\right\| _{H_{0}^{2}( \Omega )}\le 1 \}, S^k:=\left\{ u\in \mathbb {W}_{k}: \left\| u\right\| _{H_{0}^{2}( \Omega )}= 1\right\} \end{aligned}$$

and

$$\begin{aligned} S_+^{k+1}:=\left\{ u=w+se_{k+1}: \left\| u\right\| _{H_{0}^{2}( \Omega )}=1, w\in B_k, 0\le s\le 1\right\} . \end{aligned}$$

By these subspaces, we can introduce some continuous maps and minimax sequences of J as follows

$$\begin{aligned} \Lambda _k&:=\left\{ \varphi \in C(S^k, H_{0}^{2}( \Omega )): \varphi \hbox { is odd }\right\} ,\nonumber \\ \Gamma _k&:=\left\{ \varphi \in C(S_+^{k+1}, H_{0}^{2}( \Omega )): \varphi \Big |_{S^k} \in \Lambda _k\right\} , \end{aligned}$$

(2.55)

and

$$\begin{aligned} b_k:=\inf \limits _{\varphi \in \Lambda _k} \max \limits _{u \in S^k} J (\varphi (u)), \quad c_k: =\inf \limits _{h \in \Gamma _k}\max \limits _{u \in S_+^{k+1}}J (\varphi (u)), \quad k \in \mathbb {N}. \end{aligned}$$

(2.56)

For any $\delta >0$, put

$$\begin{aligned} \Gamma _k(\delta )&:=\left\{ \varphi \in \Gamma _k: J(\varphi (u)) \le b_k +\delta , u \in S^k \right\} , \end{aligned}$$

(2.57)

$$\begin{aligned} c_k(\delta )&:=\inf \limits _{h \in \Gamma _k(\delta )}\max \limits _{u \in S_+^{k+1}}J (\varphi (u)). \end{aligned}$$

(2.58)

By (2.55)–(2.58), it is obvious that $b_k \le c_k \le c_k(\delta ), k \in \mathbb {N}$. Next, we give some useful estimates for minimax values $b_k$ and $c_k(\delta )$.

Lemma 2.7

Assume that (A1), (A3), (A4), (B) are satisfied. Then for any $k\in \mathbb {N}, b_k < 0.$

Proof

Since $\mathbb {W}_{k}$ is a finite-dimensional space, by (A3), (A4), (B), for any $u \in \mathbb {W}_{k}$ we get that

$$\begin{aligned} J(u)\le \frac{1}{2} \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} + C_{14}\left\| u\right\| ^{p_2}_{H_{0}^{2}( \Omega )} +C_{15}\left\| u\right\| ^{p_3}_{H_{0}^{2}( \Omega )} +C_{16}\left\| u\right\| ^{\theta }_{H_{0}^{2}( \Omega )} -C_{17}\left\| u\right\| ^{p_1}_{H_{0}^{2}( \Omega )}. \end{aligned}$$

Hence, there exist $\varepsilon (k)>0$ and $\kappa (k)>0$ such that $J(\kappa u)< -\varepsilon , u \in S^k$. Then we set $\varphi (u)=\kappa u, u \in S^k$. By (2.56), we obtain $b_k<0$. $\square$

Lemma 2.8

Assume that (A1), (A3), (A4), (B) are satisfied. Then for any $k\in \mathbb {N}$ and any $\delta >0$, we have $c_k(\delta )<0.$

Proof

By (2.57) and (2.58), for fixed $k\in \mathbb {N}, 0<\delta <\delta '$, we have $\Gamma _k(\delta )\subset \Gamma _k(\delta ')$ and $c_k(\delta )>c_k(\delta ')$. Then we only need to prove $c_k(\delta )<0$ for any $\delta \in (0, \left| b_k\right| )$. For any $\delta \in (0, \left| b_k\right| )$, from (2.56), there exists $\varphi _0\in \Lambda _k$ such that $\max \limits _{u\in S^k}J(\varphi _0(u)) \le b_n +\frac{\delta }{2}.$ Since $\varphi _0(S^k)$ is a compact set in $H_{0}^{2}( \Omega )$, there exists a positive integer $m_0$ such that

$$\begin{aligned} \max \limits _{u\in S^k}J((P_{m_0} \circ \varphi _0)(u)) \le b_k +\delta , \end{aligned}$$

(2.59)

where $P_{m_0}$ denotes the orthogonal projective operator from $H_{0}^{2}( \Omega )$ to $\mathbb {V}_{m_0}$.

For any $c \in \mathbb {R}$, let $J^c:=\{u \in H_{0}^{2}( \Omega ): J(u)\le c\}$. Choose $\overline{\varepsilon }=-\frac{b_k+\delta }{2}>0$. By (A1), (A4), (B) and (2.13), there exists a positive constant $\rho _0$ such that if $u\in \overline{B}(0,\rho _0),$ $J(u)\le \varepsilon$, where $B(x_0,\rho )$ denotes the open ball of radius $\rho$ centered at $u_0$ in $H_{0}^{2}( \Omega )$ and $\overline{B}$ denotes the closure in $H_{0}^{2}( \Omega )$. From (2.13) and $J(0)=0$, hence $\hbox {dist}(0,J^{-\overline{\varepsilon }}) >0$. Setting

$$\begin{aligned} \rho _0':=\min \{\rho _0, \hbox {dist}(0,J^{-\overline{\varepsilon }})\}, \end{aligned}$$

then $\rho '_0>0$. By deformation theorem in [2] (or see deformation theorem in [9]), we have there exist $\varepsilon \in (0, \overline{\varepsilon })$ and a continuous map $\eta \in C([0,1]\times H_{0}^{2}( \Omega ), H_{0}^{2}( \Omega ))$ such that

$$\begin{aligned} \eta (1,u)=u, \quad \hbox {if}\quad J(u) \notin [-\overline{\varepsilon }, \overline{\varepsilon }], \end{aligned}$$

(2.60)

and

$$\begin{aligned} \eta (1,J^{\varepsilon }\backslash {B(0,{\rho '_0})})\subset J^{-\varepsilon }, \end{aligned}$$

(2.61)

where $B(0,{\rho '_0})$ is a neighborhood of $K_0$.

From (2.55), we obtain $P_{m_0} \circ \varphi _0 \in C(S^k, \mathbb {V}_{m_0})$. Since $\mathbb {V}_{k+1}$ is a metric space with the norm $\left\| \cdot \right\| _{H_{0}^{2}( \Omega )}$ and $S^k$ is a closed subset in $\mathbb {V}_{k+1}$, by Dugundji extension theorem (see Theorem 4.1 in [5]), we have there exists an extension

$$\begin{aligned} \widetilde{P_{m_0} \circ \varphi _0} : \mathbb {W}_{k+1} \rightarrow \mathbb {V}_{m_0}; \end{aligned}$$

furthermore,

$$\begin{aligned} \left( (\widetilde{P_{m_0} \circ \varphi _0})\mathbb {W}_{k+1}\right) \subset \text{ co }\left( ({P_{m_0} \circ \varphi _0})S^k \right) , \end{aligned}$$

(2.62)

where the symbol co denotes the convex hull. Since $({P_{m_0} \circ \varphi _0})S^k$ is a compact set in $\mathbb {V}_{m_0}$, by the definition of convex hull, $\hbox {co}\left( ({P_{m_0} \circ \varphi _0})S^k \right)$ is a bounded set in $\mathbb {V}_{m_0}$. Then there exists a constant $\nu$ such that

$$\begin{aligned} J(u)\le \nu , \quad u \in \hbox {co}\left( ({P_{m_0} \circ \varphi _0})S^k \right) . \end{aligned}$$

By (2.62), we have

$$\begin{aligned} J\left( (\widetilde{P_{m_0} \circ \varphi _0})u\right) \le \nu , \quad \forall u \in \mathbb {W}_{k+1}. \end{aligned}$$

(2.63)

Next, we distinguish two cases.

Case 1. $\nu \le \varepsilon$. Since $\widetilde{P_{m_0} \circ \varphi _0} \in C(\mathbb {W}_{k+1}, \mathbb {V}_{m_0})$, by (2.63), we get that

$$\begin{aligned} (\widetilde{P_{m_0} \circ \varphi _0})u \in J_{m_0}^{\varepsilon },\quad u \in \mathbb {W}_{k+1}, \end{aligned}$$

(2.64)

where $J^{\varepsilon }_{m_0}:=\{u \in \mathbb {V}_{m_0}: J(u)\le \varepsilon \}$. Define a map $\Theta$ as follows:

$$\begin{aligned} \Theta (u)= {\left\{ \begin{array}{ll} u, &{} u \notin \overline{B}(0,\rho '_0) \cap \mathbb {V}_{m_0} \\ u+\left( {\rho '}_0^2-\left\| u\right\| ^2_{H_{0}^{2}( \Omega )}\right) ^{\frac{1}{2}}e_{m_0+1}, &{}u \in \overline{B}(0,{\rho '_0}) \cap \mathbb {V}_{m_0}. \end{array}\right. } \end{aligned}$$

(2.65)

It is clear that $\Theta \in C(\mathbb {V}_{m_0}, \mathbb {V}_{m_0+1})$.

On the other hand, if $u\in \mathbb {W}_{k+1}$ and $\left\| \left( \widetilde{P_{m_0} \circ \varphi _0} \right) u\right\| _{H_{0}^{2}( \Omega )}>\rho '_0$ then by (2.64) and (2.65), we have

$$\begin{aligned}&\left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u = \left( \widetilde{P_{m_0} \circ \varphi _0} \right) u \in J^\varepsilon _{m_0},\nonumber \\&\left\| \left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u \right\| _{H_{0}^{2}( \Omega )} >\rho '_0. \end{aligned}$$

(2.66)

Otherwise, $u\in \mathbb {W}_{k+1}$ and $\left\| \left( \widetilde{P_{m_0} \circ \varphi _0} \right) u\right\| _{H_{0}^{2}( \Omega )}\le \rho '_0$ from (2.65) we have

$$\begin{aligned} \left\| \left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u \right\| _{H_{0}^{2}( \Omega )} =\rho '_0. \end{aligned}$$

(2.67)

Combining the definition of $\rho '_0$, (2.66) and (2.67), we obtain

$$\begin{aligned} \left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u \notin B(0,\rho '_0), \quad u \in \mathbb {W}_{k+1}, \end{aligned}$$

(2.68)

and

$$\begin{aligned} \left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u \in J^\varepsilon ,\quad \forall u \in \mathbb {W}_{k+1}. \end{aligned}$$

(2.69)

Define a map

$$\begin{aligned} \Theta _{m_0}: \mathbb {W}_{k+1}&\longrightarrow H_{0}^{2}( \Omega )\nonumber \\ u&\longmapsto \Theta _{m_0}(u)=\eta \left( 1,\left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u\right) . \end{aligned}$$

(2.70)

We need to prove $\Theta _{m_0}\in \Gamma _k(\delta )$ and $\max \limits _{u\in S^{k+1}_+} J(\Theta _{m_0}(u))< 0$. First, it is obvious that $\Theta _{m_0} \in C(S^{k+1}_+, H_{0}^{2}( \Omega ))$. Next, we prove $\Theta _{m_0}\Big |_{S^k}\in \Lambda _k$. By Dugundji extension theorem, we get

$$\begin{aligned} \left( \widetilde{P_{m_0}\circ \varphi _0}\right) u = \left( P_{m_0}\circ \varphi _0\right) u, \quad \forall u \in S^k. \end{aligned}$$

(2.71)

From (2.59), hence $\left( P_{m_0}\circ \varphi _0\right) u \in J^{-2\varepsilon }, u\in S^k$. By the definition of $\rho '_0$ and $J^{-2\varepsilon }\subset J^{-\varepsilon }$ implies that

$$\begin{aligned} \left\| \left( P_{m_0}\circ \varphi _0\right) u\right\| _{H_{0}^{2}( \Omega )} \ge \rho '_0, \quad \forall u \in S^k. \end{aligned}$$

(2.72)

From (2.65), (2.71) and (2.72), we have that

$$\begin{aligned} \left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u =\Theta \left( \left( {P_{m_0} \circ \varphi _0} \right) u\right) = \left( {P_{m_0} \circ \varphi _0} \right) u,\quad \forall u \in S^k. \end{aligned}$$

(2.73)

Since $\left( {P_{m_0} \circ \varphi _0} \right) u \in J^{-2\varepsilon }, \forall u \in S^k$, by (2.59), (2.60), (2.70) and (2.73), we have

$$\begin{aligned} \Theta _{m_0}(u)=\eta \left( 1,\left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u\right) = \left( {P_{m_0} \circ \varphi _0} \right) u, \quad \forall u \in S^k, \end{aligned}$$

(2.74)

which implies that $\Theta _{m_0}\Big |_{S^k}\in \Lambda _k.$ Moreover, from (2.57), (2.59) and (2.74), we have $\Theta _{m_0} \in \Gamma _k(\delta )$. Since $S^{k+1}\subset \mathbb {W}_{k+1},$ by (2.68) and (2.69), we obtain

$$\begin{aligned} \left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u \notin B(0,\rho '_0)\quad \hbox {and}\quad \left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) u\in J^\varepsilon ,\quad \forall u \in S^{k+1}. \end{aligned}$$

From (2.61) and (2.70), we get $\max \limits _{u\in S_+^{k+1}}J\left( {\Theta _{m_0}}(u)\right) \le -\varepsilon <0$ which implies that $c_k(\delta )<0.$

Case 2. $\nu > \varepsilon$. By a similar proof as in Lemmas 2.3 and 2.5, we can prove that $J\Big |_{\mathbb {V}_{m_0}} \in C^1(\mathbb {V}_{m_0}, \mathbb {R})$ and satisfies Palais–Smale condition. Moreover, $J\Big |_{\mathbb {V}_{m_0}}$ has no critical points with positive critical values on $\mathbb {V}_{m_0}$. By noncritical interval theorem (see Theorem 5.1.6 in [3]), we see that $J^{\varepsilon }_{m_0}$ is a strong deformation retract of $J_{m_0}^\nu$. So there exists a map $\psi$ such that $\psi \in C(J^{\nu }_{m_0}, J^{\varepsilon }_{m_0})$ and $\psi (u)=u$, if $u\in J^{\varepsilon }_{m_0}$ . Define a map $\Psi$ as follows:

$$\begin{aligned} \Psi : \mathbb {W}_{k+1}&\longrightarrow H_{0}^{2}( \Omega ) \nonumber \\ u&\longmapsto \Psi (u):=\psi \left( 1,\left( \Theta \circ \left( \widetilde{P_{m_0} \circ \varphi _0} \right) \right) (u)\right) . \end{aligned}$$

By a similar proof as in Case 1, we get $\Psi \in \Gamma _k(\delta )$ and $\max \limits _{u\in S_+^{k+1}}J(\Psi (u)) \le -\varepsilon <0$ which implies that $c_k(\delta )<0.$ The proof of Lemma 2.8 is complete. $\square$

Lemma 2.9

Suppose that f satisfies (A1), (A3), (A4) and g satisfies (B). Then there exists a positive constant $C_{18}$ independent of k such that for all k large enough

$$\begin{aligned} b_k \ge -C_{18}k^{\frac{-4p}{N (2-p)}}. \end{aligned}$$

(2.75)

Proof

For any $\varphi \in \Lambda _k (k\ge 2)$ when $0\notin \varphi (S^k)$, then the genus $\gamma (\varphi (S^k))$ is well defined and $\gamma (\varphi (S^k)) \ge \varphi (S^k) =k.$ By Proposition 7.8 in [12], hence $\varphi (S^k) \cap \mathbb {V}^\bot _{k-1}\ne \emptyset .$ Otherwise, if $0\in \varphi (S^k)$ then $0\in \varphi (S^k) \cap \mathbb {V}^\bot _{k-1}.$ So for any $\varphi \in \Lambda _k (k\ge 2)$ we have $\varphi (S^k) \cap \mathbb {V}^\bot _{k-1} \ne \emptyset .$ Therefore, for any $\varphi \in \Lambda _k (k\ge 2)$, we obtain

$$\begin{aligned} \max \limits _{u\in S^k}J(\varphi (u)) \ge \inf \limits _{u\in \mathbb {V}^\bot _{k-1}}J(u). \end{aligned}$$

(2.76)

From (A1), (A4), (B), (2.10) and (2.13), we get that

$$\begin{aligned} J(u)&\ge \frac{1}{2} \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} -C_{19}\left\| u\right\| ^p_{L^p(\Omega )} - \pi (u) \int \limits _{\Omega }F_2(x,u)\mathrm {d}x -\psi (u)\nonumber \\&\ge \frac{1}{4} \left\| u\right\| ^2_{H_{0}^{2}( \Omega )} -C_{19}\left\| u\right\| ^p_{L^2(\Omega )} , \forall u \in H_{0}^{2}( \Omega ). \end{aligned}$$

(2.77)

Moreover, by $u \in \mathbb {V}_{k-1}^\bot ,$ hence

$$\begin{aligned} \left\| u\right\| _{L^2(\Omega )} \le \lambda _k^{\frac{-1}{2}}\left\| u\right\| _{H_{0}^{2}( \Omega ))}. \end{aligned}$$

(2.78)

Combining (2.56), (2.76), (2.77) and (2.78), for any $k\ge 2$, we have

$$\begin{aligned} b_k \ge \inf \limits _{t\ge 0} \left( \frac{1}{4} t^2 -C_{19}\lambda _k^{\frac{-p}{2}}t^p\right) =-C_{20}\lambda _k^{\frac{-p}{2-p}}, \end{aligned}$$

(2.79)

where $C_{20}$ is a positive constant independent of k and $\lambda _k$. On the other hand, it follows from Agmon’s generalization [1] of Weyl’s formula [15], which in fact is an extension of earlier work of Pleijel [10] for $N=2$, we have

$$\begin{aligned} \lambda _k \ge C_{21} k^{\frac{4}{N}}. \end{aligned}$$

(2.80)

Combining (2.79) and (2.80), we arrive at the conclusion of the lemma. $\square$

Lemma 2.10

Suppose that $c_k=b_k$ for $k\ge k_0,$ where $k_0\in \mathbb {N}.$ Then there exists a positive integer $k_1$ such that

$$\begin{aligned} \left| b_k\right| \ge C_{22}k^{\frac{2}{2-\theta }}, \quad k \ge k_1, \end{aligned}$$

(2.81)

where $C_{22}$ is a positive constant independent of k.

Proof

For any $k\ge k_0$ and any $\varepsilon \in (0, \left| b_k\right| )$, by (2.56) there exists a map $\varphi _0\in \Gamma _k$ such that

$$\begin{aligned} \max \limits _{u\in S^{k+1}_+} J(\varphi _0(u)) < c_k +\varepsilon =b_k+\varepsilon . \end{aligned}$$

(2.82)

From $S^{k+1}=S^{k+1}_+ \cup (-S^{k+1}_+), \varphi _0$ can be continuously extended to $S^{k+1}$ as an odd function, also denoted by $\varphi _0$, then $\varphi _0\in \Lambda _{k+1}$. From (2.56), we have

$$\begin{aligned} b_{k+1}\le \max \limits _{u\in S^{k+1}} J(\varphi _0(u)) =J(\varphi _0(u_0)) \end{aligned}$$

(2.83)

for some $u_0\in S^{k+1}$. If $u_0\in S^{k+1}_+$, in combination with (2.27), (2.82) and (2.83), we have

$$\begin{aligned} b_{k+1}< b_k+\varepsilon +C_8 \left| b_{k+1}\right| ^{\frac{\theta }{2}}. \end{aligned}$$

(2.84)

Otherwise, $u_0\in -S^{k+1}_+$, from (2.27) and (2.82), we get that

$$\begin{aligned} J(\varphi _0(u_0))&\le J(\varphi _0(-u_0)) +C_8\left| J(\varphi _0(u_0))\right| ^{\frac{\theta }{2}}\nonumber \\&\le b_{k}+\varepsilon + C_8\left| J(\varphi _0(u_0))\right| ^{\frac{\theta }{2}}. \end{aligned}$$

(2.85)

Next, we consider two possible cases.

Case 1. $J(\varphi _0(u_0)) \le \left| b_{k+1}\right|$, from (2.83) and (2.84), we obtain

$$\begin{aligned} b_{k+1}< b_k+\varepsilon +C_8 \left| b_{k+1}\right| ^{\frac{\theta }{2}}. \end{aligned}$$

(2.86)

Case 2. $J(\varphi _0(u_0)) > \left| b_{k+1}\right|$. By (2.82), there exists $u_1\in S^{k+1}_+$ such that

$$\begin{aligned} J(\varphi _0(u_1))<b_k+\varepsilon <0. \end{aligned}$$

(2.87)

Since $J\circ \varphi _0 \in C(S^{k+1}, \mathbb {R})$ and $S^{k+1}$ is a connected space with the norm $\left\| \cdot \right\| _{H_{0}^{2}( \Omega ))},$ by the intermediate value theorem, there exists $u_2\in S^{k+1}$ such that $J(\varphi _0(u_2)) =\frac{\left| b_{k+1}\right| }{2}.$ By (2.82), hence $u_2\in -S^{k+1}$. From (2.27) and (2.82), we get

$$\begin{aligned} J(\varphi _0(u_2))&\le J(\varphi _0(-u_2)) +C_8 \left| J(\varphi _0(u_2)) \right| ^{\frac{\theta }{2}} \\&\le b_k +C_8 \left| J(\varphi _0(u_2)) \right| ^{\frac{\theta }{2}}, \end{aligned}$$

which implies that

$$\begin{aligned} b_{k+1}\le b_k +\varepsilon +C_8 \left| b_{k+1}\right| ^{\frac{\theta }{2}}. \end{aligned}$$

(2.88)

By Lemma 2.7, (2.84), (2.86) and (2.88), it is easy to see that

$$\begin{aligned} \left| b_k \right| \le \left| b_{k+1} \right| +C_8 \left| b_{k+1}\right| ^{\frac{\theta }{2}}, \quad k \ge k_0. \end{aligned}$$

(2.89)

Next, we show that (2.89) implies (2.81). The proof will be done by induction. First, we introduce a useful inequality as follows:

$$\begin{aligned} \left( 1+t\right) ^\alpha \ge 1 +\frac{\alpha t}{2}, \quad t \in [0, \beta ], \end{aligned}$$

(2.90)

where $\alpha , \beta$ are positive constants and $\beta$ depends on $\alpha$. Set $\alpha =2(\theta -2)^{-1}$. In view of (2.90), there exists $\widetilde{k_0}\in \mathbb {N}$ such that

$$\begin{aligned} \left( 1+\frac{1}{k}\right) ^{\frac{2}{\theta -2}} \ge 1 +\frac{1}{(\theta -2)k}, \quad k \ge \widetilde{k_0}. \end{aligned}$$

(2.91)

Define

$$\begin{aligned} C_{22}:=\min \left\{ k_1^{\frac{2}{\theta -2}}\left| b_{k_1}\right| , \left( \frac{1}{C_8(\theta -2)}\right) ^{\frac{2}{\theta -2}} \right\} , \end{aligned}$$

(2.92)

where $k_1=\max \{k_0, \widetilde{k_0}\}.$ Then we claim (2.81) holds. By (2.92), we have

$$\begin{aligned} \left| b_{k_1}\right| \ge C_{22}k_1^{\frac{2}{2-\theta }}. \end{aligned}$$

(2.93)

Suppose that (2.81) holds for $k \ge k_1$. Then we only need to prove (2.81) also holds for $k+1$. If not, we get that

$$\begin{aligned} \left| b_{k+1}\right| \le C_{22}(k+1)^{\frac{2}{2-\theta }}. \end{aligned}$$

(2.94)

Since (2.81) holds for k, by (2.27), (2.89) and (2.94), we obtain

$$\begin{aligned} C_{22}k_1^{\frac{2}{2-\theta }}&\le \left| b_k\right| \le \left| b_{k+1} \right| +C_8 \left| b_{k+1}\right| ^{\frac{\theta }{2}} \le C_{22}(k+1)^{\frac{2}{2-\theta }} \nonumber \\&\quad +C_8 C_{22}^{\frac{\theta }{2}} (k+1)^{\frac{\theta }{2-\theta }}. \end{aligned}$$

(2.95)

When we divide (2.95) by $C_{22} (k+1)^{\frac{2}{2-\theta }}$ on both sides, in view of (2.92), we get that

$$\begin{aligned} \left( 1+\frac{1}{k}\right) ^{\frac{2}{\theta -2}}< 1+C_8 C_{22}^{\frac{\theta -2}{2}} \frac{1}{k+1} < 1+C_8 C_{22}^{\frac{\theta -2}{2}} \frac{1}{k} \le 1 +\frac{1}{(\theta -2)k}, \end{aligned}$$

which contradicts (2.91). So (2.81) holds. The proof of Lemma 2.10 is complete. $\square$

Lemma 2.11

Suppose that $c_k>b_k$. Then for any $\delta \in (0, c_k-b_k), c_k(\delta )$ given by (2.57) is a critical value of J.

Proof

By using deformation theorem in [2], the proof of this lemma is similar to the one of Lemma 1.57 in [11]. We omit the details. $\square$

Proof of Theorem 1.1 From (1.4), Lemmas 2.7, 2.9 and 2.10, it is impossible that $c_k=b_k$ for all large k, we can choose subsequence $\{k_j\}_{j=1}^\infty \subset \mathbb N$ such that $c_{k_j} > b_{k_j}$. By Lemmas 2.8, 2.9 and 2.11, there exists a sequence of critical points $\{u_{k_j}\}_{j=1}^\infty$ of J such that

$$\begin{aligned} -C_{18}k_j^{\frac{-2p}{N (2-p)}} \le b_{k_j}< c_{k_j} \le c_{k_j}(\delta _j) =J(u_{k_j})< 0, \end{aligned}$$

(2.96)

where $\delta _j \in (0, c_{k_j}-b_{k_j})$. It is obvious that $u_{k_j} \ne 0, j \in \mathbb {N}$. Next, we consider the following two possible cases.

Case 1. $\left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} >2T_0$. From (2.2), (2.3) and (2.16), hence

$$\begin{aligned} \pi (u_{k_j}) =1\quad \hbox {and}\quad \psi '(u_{k_j}) =0. \end{aligned}$$

By (A2), (2.5) and (2.28), we get that

$$\begin{aligned} \overline{I}(u_{k_j})&= \overline{I}(u_{k_j}) -\mu ^{-1} \langle \widetilde{I}'(u_{k_j}), u_{k_j}\rangle \nonumber \\&= 2 A \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} +\int \limits _{\Omega } \left( \mu ^{-1}f_1(x,u_{n_j})u_{n_j}-F_1(x,u_{n_j}) \right) \mathrm {d}x \nonumber \\&\le A \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))}. \end{aligned}$$

(2.97)

Case 2. $\left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} \le 2T_0$. By Lemmas 2.1, 2.3, (A2), (A4), (B) (2.5) and (2.28), we get that

$$\begin{aligned}&\overline{I}(u_{k_j}) \le \frac{1}{2} \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} -\int \limits _{\Omega }F_1(x,u_{k_j})\mathrm {d}x + C_3 C_{p_3}^{p_3} \left\| u_{k_j}\right\| ^{p_3}_{H_{0}^{2}( \Omega ))}\\&+ C_4 C_{\theta }^{\theta }\left\| u_{k_j}\right\| ^{\theta }_{H_{0}^{2}( \Omega ))}, \end{aligned}$$

and

$$\begin{aligned} \langle \widetilde{I}'(u_{k_j}), u_{k_j}\rangle&\ge \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} -\int \limits _{\Omega }f_1(x,u_{k_j})u_{k_j}\mathrm {d}x - 9 C_3 C_{p_3}^{p_3} \left\| u_{k_j}\right\| ^{p_3}_{H_{0}^{2}( \Omega ))} \\&\quad -89C_4 C_{\theta }^{\theta }\left\| u_{k_j}\right\| ^{\theta }_{H_{0}^{2}( \Omega ))}. \end{aligned}$$

Hence,

$$\begin{aligned} \overline{I}(u_{k_j})&= \overline{I}(u_{k_j}) -\mu ^{-1} \langle \widetilde{I}'(u_{k_j}), u_{k_j}\rangle \nonumber \\&\le 2 A \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} + \frac{\mu +9}{\mu } C_3 C_{p_3}^{p_3} \left\| u_{k_j}\right\| ^{p_3}_{H_{0}^{2}( \Omega ))} +\frac{\mu +89}{\mu }C_4 C_{\theta }^{\theta } \left\| u_{k_j}\right\| ^{\theta }_{H_{0}^{2}( \Omega ))}\nonumber \\&\le A \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))}. \end{aligned}$$

(2.98)

In both cases, by (2.11), (2.97) or (2.98), we get $\ell (u_{k_j}) =0$ and $\ell '(u_{k_j}) =0.$ Hence,

$$\begin{aligned} J(u_{k_j}) =\overline{I}(u_{k_j}) \le A \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} < 0. \end{aligned}$$

By (2.96), it is easy to see that

$$\begin{aligned} \left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} \rightarrow 0 \hbox { as } j \rightarrow \infty . \end{aligned}$$

So there exists $j_0 \in \mathbb {N}$ such that $\left\| u_{k_j}\right\| ^2_{H_{0}^{2}( \Omega ))} < T_0$ for all $j \ge j_0$. By (2.3) and (2.16), hence

$$\begin{aligned} \pi (u_{k_j}) =1,\quad \pi '(u_{k_j}) =0\quad \hbox {for all}~j \ge j_0. \end{aligned}$$

In combination with (2.52), (2.16) and (2.28), when j is large enough, we conclude that $u_{k_j}$ are weak solutions of the problem (1.2). $\square$