In this paper, we shall investigate the existence of weak solutions for the fourth order elliptic problem

$\label{pro}\begin{cases} \Delta^2 u+c \Delta u=f(x,u) \ \text{ in } \Omega,\\ u=\Delta u=0 \ \text{ on } \partial\Omega, \end{cases}$

(1)

where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, the parameter $c$ is less than the first eigenvalue of $(-\Delta,H_0^1)$ and $f\in C(\overline{\Omega}\times\mathbb{R}, \mathbb{R})$.

It is well-known that fourth order problems were studied by many authors. For example, $u^{(4)}(t)=f(t,u(t))$ subject to boundary value conditions $u(0) = u(1) = u''(0) = u''(1) = 0$ can be used to model the deflection of elastic beams simply supported at the endpoints [1-3]. In [4], Lazer and Mckenna pointed out that fourth order elliptic problems furnish a model to study traveling waves in suspension bridges. Since then, more general nonlinear fourth order elliptic boundary value problems were studied, we refer the interested reader to [5-13].

Meanwhile, as is known to all, fountain and dual fountain theorems by Bartsch and Willem [14, 15] are effective tools for studying the existence of infinitely many large energy solutions and small energy solutions, for instance, see [16-19] and the references therein.

In [5], Yang and Zhang consider the existence of positive, negative and sign-changing solutions for (1.1). They present their results on invariant sets of the gradient flows of the corresponding variational functionals.

Chen and Tang [16] investigated the fractional boundary value problem of the following form

$\left\{ \begin{array}{l} \frac{d}{{dt}}\left( {\frac{1}{2}{\;_0}D_t^{ - \beta }u'(t) + \frac{1}{2}{\;_t}D_T^{ - \beta }u'(t)} \right) + \nabla F(t,u(t)) = 0\;{\rm{a}}{\rm{.e}}{\rm{. }}t \in [0,T],\\ u(0) = u(T) = 0. \end{array} \right.$

In their paper, they adopted fountain and dual fountain theorems to obtain the existence of infinitely many solutions under some adequate conditions. It is no doubt that the results in the literature are significantly improved.

In this paper, we first adopt mountain pass theorem to obtain the existence of nontrivial solutions for (1.1) under some appropriate conditions imposed on $f$, i.e., sublinearity at $0$ and superlinearity at $\infty$ with respect to $u$. Secondly, we only consider the nonlinearity $f$ is asymptotically linear at infinity and we utilize fountain theorem to obtain the existence of infinitely many solutions for (1.1). Note that, this condition at infinity is indeed $(f_1)$ of [5]. In [5], the authors assume that the nonlinearity $f$ satisfies conditions $(f_1), (f_4), (f_6)$ and $(f_7)$ (see [5, P130]) to obtain the existence of weak solutions for (1.1). However, in this paper, we only need conditions $(f_1)$ and $(f_7)$ to establish the existence of infinitely many solutions for (1.1). Therefore, our results here improve and extend the corresponding ones in [5].

Let $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$. We will always assume the parameter $c$ is less than the first eigenvalue of $(-\Delta,H_0^1)$. Denote by $0 < \lambda_{1} < \lambda_{2} \leq \dots \leq \lambda_{j} \leq \dots$ the eigenvalues of $(-\Delta,H_0^1)$ and $\mu_k(c)=\lambda_k(\lambda_k-c)$ the eigenvalues of $(\Delta^2+c\Delta,H_0^1\cap H^2)$. We also denote by $\varphi_j$ the eigenfunction associated with $\lambda_j$ and consequently with $\mu_j$, and $\varphi_1>0$ for $x\in \Omega$. We define a space $X:=H_0^1\cap H^2$. Clearly, $X$ is a Hilbert space with the inner product

$ (u,v)=\int_{\Omega}(\Delta u \Delta v-c\nabla u\nabla v) dx. $

We denote by $\|u\|_2$ the norm in $L^2(\Omega)$ and $\|u\|$ the norm in $X$ which is given by $\|u\|^2=(u,u)$. Furthermore, we have the Poincaréinequality $\|u\|^2\ge \mu_1 \|u\|_2^2$.

We define a functional on $X$ as follows

$\label{J}J(u)=\frac{1}{2}\int_\Omega (|\Delta u|^2-c|\nabla u|^2) dx-\int_\Omega F(x,u)dx= \frac{1}{2}\|u\|^2-\int_\Omega F(x,u)dx,$

(2.1)

where $F(x,u)=\int_0^u f(x,t)dt$. Clearly, the existence of weak solutions for (1.1) is equivalent to the existence of critical points of the functional $J$. By simple computation, we have

$\label{Jprime}(J'(u),v)=\int_{\Omega}(\Delta u \Delta v-c\nabla u\nabla v) dx -\int_\Omega f(x,u)vdx=(u,v)-(Au,v),\ \forall u,v\in X,$

(2.2)

where $(Au,v)=\int_\Omega f(x,u)vdx,\ \forall u,v\in X$. Obviously, $J \in C^1(X, \mathbb{R})$ (see line 7 from below in [13, P. 798]).

As we have mentioned, we utilize the critical point theory to prove our main results. Let us recall two definitions about (PS) condition and (PS)$_c$ condition that will be used below. One can refer to [20-23] for more details.

Definition 2.1 Let $X$ be a real Banach space and $J\in C^1(X,\mathbb{R})$. We say $J$ satisfies (PS) condition if for every sequence $\{u_n\}\subset X$ such that $J(u_n)$ is bounded and $J'(u_n)\to 0$ as $n\to \infty$, there exists a subsequence of $\{u_n\}$ which is convergent in $X$.

Definition 2.2 Let $X$ be a real Banach space, $J\in C^1(X,\mathbb{R})$ and $d\in \mathbb{R}$. We say $J$ satisfies (PS)$_c$ condition if the existence of a sequence $\{u_n\}\subset X$ such that $J(u_n)\to d$ and $J'(u_n)\to 0$ as $n\to \infty$ lead to $d$ is a critical value of $J$.

Lemma 2.1 (Mountain Pass Theorem) Let $X$ be a Banach space and $J\in C^1(X,\mathbb{R})$ be a functional satisfying (PS) condition. If $e\in X$ and $0<r<\|e\|$ are such that

$a:=\max\{J(0),J(e)\}<\mathop {\inf }\limits_{\left\| u \right\| = r} J(u)=:b,$

then $c:=\inf_{\gamma\in \Gamma}\sup_{t\in [0,1]}J(\gamma(t))$ is a critical value of $J$ with $c\ge b$, where $\Gamma$ is the set of paths joining the points $0$ and $e$, i.e., $\Gamma:=\{\gamma\in C([0,1],X):\gamma(0)=0,\ \gamma(1)=e\}$.

For $j,k\in \mathbb{N}$, $X=\overline{\text{span}}\{\varphi_j:j=1,2\ldots\}$. Denote $X_j:=\text{span} \{\varphi_j\}$, $Y_k:=\bigoplus_{j=1}^kX_j$ and $Z_k:=\overline{\bigoplus_{j=k+1}^\infty X_j}$. Clearly, $X=\overline{\bigoplus_{j\in \mathbb{N}}X_j}$ with dim$X_j<\infty$ for all $j\in \mathbb{N}$. We also find $X=Y_k\bigoplus Z_k$.

Lemma 2.2 (see [14]) Let $X$ be defined above. Suppose that

(A1) $J\in C^1(X,\mathbb{R})$ is an even functional. If for every $k\in \mathbb{N}$, there exist $\rho_k>r_k>0$ such that

(A2) $a_k:=\max\limits_{u\in Y_k, \|u\|=\rho_k}J(u)\le 0$;

(A3) $b_k:=\inf\limits_{u\in Z_k, \|u\|=r_k}J(u)\to \infty$ as $k\to \infty$;

(A4) $J$ satisfies (PS)$_c$ condition for all $c>0$, then $J$ has an unbounded sequence of critical values.

Theorem 2.1 If $f$ satisfies the following two conditions:

(H1) $f(x,0)=0$;

(H2) $\lim\limits_{t\to 0}\frac{f(x,t)}{t}=\xi$, $\lim\limits_{|t|\to \infty}\frac{f(x,t)}{t}=\eta$, uniformly a.e. in $x\in \Omega$, where

$0\le \xi<\mu_1=\lambda_1(\lambda_1-c)<\eta<+\infty,$

then (1.1) has at least a weak solution.

Lemma 2.3 Suppose (H1), (H2) hold, then $J$ satisfies (PS) condition.

Proof Since $\Omega$ is bounded and (H2) holds, then if $\{u_k\}$ is bounded in $X$, by using the Sobolev embedding theorem and the standard procedures, we can get a subsequence converges strongly in $X$. So we need only to show that $\{u_k\}$ is bounded in $X$.

Assume that $\{u_k\}\subset X$ is a (PS) sequence, i.e.,

$J(u_k)\to d, \quad J'(u_k)\to 0, \quad \text{as } k\to \infty.$

(2.3)

From (H2) we know that

$|f(x,t)t|\le C(1+|t|^2).$

(2.4)

(2.3) implies that for all $\varphi\in X$,

$(J'(u_k),\varphi)= \int_{\Omega}(\Delta u_k \Delta \varphi-c\nabla u_k\nabla \varphi) dx -\int_\Omega f(x,u_k)\varphi dx\to 0.$

(2.5)

Setting $\varphi=u_k$ and using (2.4), we have

$\|u_k\|^2= \int_\Omega f(x,u_k)u_kdx+(J'(u_k),u_k)\le C|\Omega|+C\|u_k\|_2^2+o(1)\|u_k\|.$

(2.6)

We claim that $\|u_k\|_2$ is bounded. Assume, by contradiction, that passing to a subsequence, $\|u_k\|_2^2\to \infty$ as $k\to \infty$. We put $\omega_k:=\frac{u_k}{\|u_k\|_2}$, then $\|\omega_k\|_2=1$. Moreover, from (2.6) we know

$\|\omega_k\|^2\le o(1)+C+\frac{o(1)}{\|u_k\|_2}\frac{\|u_k\|}{\|u_k\|_2}\le o(1)+C+o(1)\|\omega_k\|.$

(2.7)

Hence, $\|\omega_k\|$ is bounded. Passing to a subsequence, we may assume that there exists $\omega\in X$ and $\|\omega\|_2=1$ such that

$\omega_k\rightharpoonup w, \text{ weakly in } X, k\to \infty, \quad \omega_k\to w, \text{ strongly in } L^2(\Omega), k\to \infty.$

From (2.5) we derive

$\int_{\Omega}(\Delta \omega \Delta \varphi-c\nabla \omega\nabla \varphi) dx -\int_\Omega \eta\omega\varphi dx= 0, \ \forall \varphi\in X.$

(2.8)

Then $\omega\in X$ is a weak solution of the equation

$\Delta^2 \omega+c \Delta \omega=\eta \omega.$

Taking $\varphi(x) = \varphi_1(x)$, from (2.8) we have

$\int_{\Omega}(\Delta \omega \Delta \varphi_1-c\nabla \omega\nabla \varphi_1) dx -\int_\Omega \eta\omega\varphi_1dx= 0.$

(2.9)

On the other hand, since $\varphi_1(x)> 0$ is the eigenfunction of $\lambda_1(\lambda_1-c)$, we have also

$\int_{\Omega}(\Delta \omega \Delta \varphi_1-c\nabla \omega\nabla \varphi_1) dx -\int_\Omega \lambda_1(\lambda_1-c)\omega\varphi_1dx= 0.$

(2.10)

Together (2.9) with (2.10), note that $\|\omega\|_2=1$, we know that $\lambda_1(\lambda_1-c)=\eta$, which contradicts $\lambda_1(\lambda_1-c)<\eta$. Hence $\|u_k\|_2$ is bounded. Then, from (2.6) we know that $\{u_k\}$ is bounded in $X$. This completes the proof.

Proof of Theorem 2.1 (H2) implies that, for any $\varepsilon > 0$, there exists $C_1 > 0$, such that

$F(x,t)\ge \frac{1}{2}(\eta-\varepsilon)t^2-C_1, \forall x\in \Omega, t\not =0.$

(2.11)

Taking $\varepsilon > 0$ such that $\eta -\varepsilon >\mu_1$, $\phi =\varphi_1$, from (2.11) we obtain

$ J(s\phi)\le \frac{1}{2}\int_\Omega (|\Delta (s\phi)|^2-c|\nabla (s\phi)|^2) dx- \frac{1}{2}(\eta-\varepsilon)\int_\Omega s^2\phi^2dx + C_1 |\Omega|\\ \le \frac{s^2}{2}\|\phi\|^2-\frac{s^2}{2}(\eta-\varepsilon)\|\phi\|^2_2+ C_1 |\Omega|\\ \le \frac{s^2}{2}\left(1-\frac{\eta-\varepsilon}{\mu_1}\right)\|\phi\|^2+ C_1 |\Omega|. $

Therefore, by $1-\frac{\eta-\varepsilon}{\mu_1}<0$ implies

$\mathop {\lim }\limits_{s \to \infty } J(s\phi ) \to - \infty .$

(2.12)

From (H2), we can find $\alpha$, such that $2 < \alpha < 2^*$, where $2^*=\begin{cases} \frac{2N}{N-2}, N>2,\\ +\infty, N\le 2. \end{cases}$ (H1), (H2) imply that for all given $\varepsilon > 0$, there exists $C_0 > 0$, such that

$F(x,t)\le \frac{1}{2}(\xi+\varepsilon)|t|^2+C_0 |t|^\alpha .$

(2.13)

(2.13), the Poincaré inequality and the Sobolev embedding theorem enable us to obtain

$J(u) \ge \frac{1}{2}\|u\|^2-\frac{\xi+\varepsilon}{2}\int_\Omega |u|^2dx-C_0 \int_\Omega |u|^\alpha dx \ge \left(\frac{1}{2}-\frac{\xi+\varepsilon}{2\mu_1}\right)\|u\|^2-C_s \|u\|^\alpha,$

(2.14)

where $C_s$ is a constant. In (2.14), by taking $\varepsilon>0$ such that $\xi+\varepsilon<\mu_1$, and choosing $\|u\|= \rho > 0$ small enough, we obtain $J(u) \ge R > 0$, if $\|u\|= \rho$.

From (2.12), we know that there exists $e \in X$, $\|e\| > \rho$, such that $J(e)<0$. Define

$\Gamma:=\{\gamma: [0,1]\to X| \gamma \text{ is continuous and } \gamma (0) = 0, \gamma (1) = e\},$

and $c=\inf_{\gamma\in \Gamma}\max_{t\in [0,1]}J(\gamma(t))$. From Lemma 2.1 it follows that

$J(0)=0, J(e)<0, \text{ and } J(u)|_{\partial B_\rho}\ge R>0. $

Moreover, $J$ satisfies (PS) condition by Lemma 2.3. By the mountain pass theorem, we know $c$ is a critical value of $J$ and there is at least one nontrivial critical point in $X$ corresponding to this value. This completes the proof.

Theorem 2.2 If $f$ satisfies the following two conditions:

(H3) $ \mu_k < \liminf\limits_{|t|\to \infty}\frac{f(x,t)}{t}\le \limsup\limits_{|t|\to \infty}\frac{f(x,t)}{t}< \mu_{k+1}, \text{ uniformly in } \Omega; $

(H4) $f(t,-u)+f(t,u)=0$, then (1.1) has infinitely many weak solutions.

Lemma 2.4 (see [5, Lemma 4.1]{12}) Let $f$ satisfy (H3). Then $J$ satisfies (PS) condition.

Proof Take $\mu_k < b_1 \le b_2 < \mu_{k+1} $ and $M>0$ such that for $|u|\ge M$, $b_1\le \frac{f(x,u)}{u}\le b_2$ by (H3). Now let $\{u_k\}$ be a (PS) sequence for $J(u)$. Writing $u_k=v_k+w_k$ with $v_k\in Y_k$ and $w_k\in Z_k$. Considering the inner product of $J'(u)$ and $v_k-w_k$, we find

$o(1)\cdot \|u_k\| =(J'(u_k),v_k-w_k)=(u_k,v_k-w_k)-\int_\Omega f(x,u_k)(v_k-w_k)dx\\ =(v_k+w_k,v_k-w_k)-\int_{|u_k|\ge M} \frac{f(x,u_k)}{u_k}(v^2_k-w^2_k)dx-\int_{|u_k|<M} f(x,u_k)(v_k-w_k)dx\\ \le \|v_k\|^2-\|w_k\|^2-b_1 \int_{|u_k|\ge M}v_k^2 dx+b_2 \int_{|u_k|\ge M}w_k^2 dx-\int_{|u_k|< M} f(x,u_k)(v_k-w_k)dx\\ = \|v_k\|^2-\|w_k\|^2-b_1 \int_{\Omega}v_k^2 dx+b_1 \int_{|u_k|< M}v_k^2 dx+b_2 \int_{\Omega}w_k^2 dx-b_2 \int_{|u_k|< M}w_k^2 dx\\ -\int_{|u_k|< M} f(x,u_k)(v_k-w_k)dx\\ \le \|v_k\|^2-\|w_k\|^2-\frac{b_1}{\mu_k} \|v_k\|^2+\frac{b_2}{\mu_{k+1}} \|w_k\|^2+b_1 \int_{|u_k|< M}v_k^2 dx -b_2 \int_{|u_k|< M}w_k^2 dx\\ -\int_{|u_k|< M} f(x,u_k)(v_k-w_k)dx.$

(2.15)

By (2.15) and Hölder inequality, we obtain

$ o(1)\cdot \|u_k\|\le \left(1-\frac{b_1}{\mu_k}\right)\|v_k\|^2+\left(\frac{b_2}{\mu_{k+1}}-1\right)\|w_k\|^2+\frac{b_1b_2}{b_2-b_1} \int_{|u_k|< M}u_k^2 dx\\ +\left(\int_{|u_k|< M}|f(x,u_k)|^2dx\right)^{\frac{1}{2}}\left(\int_{|u_k|< M}|v_k-w_k|^2dx\right)^{\frac{1}{2}}\\ \le \left(1-\frac{b_1}{\mu_k}\right)\|v_k\|^2+\left(\frac{b_2}{\mu_{k+1}}-1\right)\|w_k\|^2+\frac{b_1b_2}{b_2-b_1} M^2 |\Omega| +C\left(\int_{|u_k|< M}|u_k|^2dx\right)^{\frac{1}{2}} \\ \le \left(1-\frac{b_1}{\mu_k}\right)\|v_k\|^2+\left(\frac{b_2}{\mu_{k+1}}-1\right)\|w_k\|^2+\frac{b_1b_2}{b_2-b_1} M^2 |\Omega| +CM\sqrt{|\Omega|}\\ \le -a \|u_k\|^2+\frac{b_1b_2}{b_2-b_1} M^2 |\Omega| +CM\sqrt{|\Omega|}.$

So, $\{u_k\}$ is bounded, where $a = \min\left\{ \frac{b_1}{\mu_k}-1,1 - \frac{b_2}{\mu_{k+1}} \right\} > 0$. A standard argument shows that $J (u)$ satisfies e (PS) condition. This completes the proof.

Proof of Theorem 2.2 (H4) and Lemma 2.4 enable us to obtain that (A1) and (A4) in Lemma 2.2 are satisfied.

By $\liminf\limits_{|t|\to \infty}\frac{f(x,t)}{t}>\mu_k$, there exist $M_1>0$ and $\varepsilon>0$ such that $f(x,t)\ge (\mu_k+\varepsilon) t $, for all $|t|\ge M_1$ and $x\in \Omega$. We know $f(x,t)-(\mu_k+\varepsilon) t$ is continuous and bounded on $x\in \Omega$ and $|t|\le M_1$, and thus there exists $C>0$ such that $-C\le f(x,t)-(\mu_k+\varepsilon) t\le C$. Therefore,

$f(x,t)\ge (\mu_k+\varepsilon)t-C, \ \forall (x,t)\in \Omega\times \mathbb{R}.$

(2.16)

By the definition of $F(x,u)$, we see

$F(x,u)=\int_0^u f(x,t)dt\ge \int_0^u [(\mu_k+\varepsilon)t-C]dt\ge \frac{u^2}{2}(\mu_k+\varepsilon)-Cu, \ \forall (x,t)\in \Omega\times \mathbb{R}.$

(2.17)

For any $u \in Y_k$, and it is easy to verify that $\|\cdot\|_2$ is a norm of $Y_k$. Since all the norms of a finite dimensional normed space are equivalent, so there exists positive constant $C_1$ such that $\|u\|_2\le C_1\|u\|$. In view of (2.17), by Hölder inequality, we obtain

$J(u) =\frac{1}{2}\|u\|^2-\int_\Omega F(x,u)dx\le \frac{1}{2}\|u\|^2-\int_\Omega \frac{u^2}{2}(\mu_k+\varepsilon)dx+C|\Omega|^{\frac{1}{2}}\|u\|_2\\ \le \frac{1}{2}\|u\|^2-\frac{\mu_k+\varepsilon}{2\mu_k}\|u\|^2+CC_1|\Omega|^{\frac{1}{2}}\|u\|=\frac{1}{2}\|u\|^2 \left(1-\frac{\mu_k+\varepsilon}{\mu_k}\right)+CC_1|\Omega|^{\frac{1}{2}}\|u\|. $

(2.18)

Since $1-\frac{\mu_k+\varepsilon}{\mu_k}<0$, then there exists positive constants $d_k$ such that

$J(u)\le 0, \text{ for each } u\in Y_k \text{ and } \|u\|\ge d_k.$

(2.19)

On the other hand, by $\limsup\limits_{|t|\to \infty}\frac{f(x,t)}{t}<\mu_{k+1}$, there exist $M_2>0$ and $\varepsilon\in (0,\mu_{k+1})$ such that

$f(x,t)\le (\mu_{k+1}-\varepsilon)t, \ |t|\ge M_2$

and $x\in \Omega$. For the reason that $f(x,t)- (\mu_{k+1}-\varepsilon)t$ is continuous and bounded on $|t|\le M_2$ and $x\in \Omega$, then there is a $C>0$ such that $f(x,t)- (\mu_{k+1}-\varepsilon)t\le C,\ |t|\le M_2$ and $x\in \Omega$. Consequently,

$f(x,t)\le (\mu_{k+1}-\varepsilon)t+ C, \ \forall (x,t)\in \Omega\times \mathbb{R}.$

(2.20)

Therefore, we have

$F(x,u)\le \frac{u^2}{2}(\mu_{k+1}-\varepsilon)+Cu, \ \forall (x,t)\in \Omega\times \mathbb{R}.$

(2.21)

For any $u \in Z_k$, let $ \beta_k:=\sup\limits_{u\in Z_k, \|u\|=1}\|u\|_2$. Since $X$ is compactly embedded into $L^2(\Omega)$, there holds (see [21, Lemma 3.8]), $\beta_k\to 0, \text{ as } k\to \infty$. By (2.21) and Hölder inequality, we arrive at

$J(u) =\frac{1}{2}\|u\|^2-\int_\Omega F(x,u)dx \ge \frac{1}{2}\|u\|^2 - \int_\Omega \frac{u^2}{2}(\mu_{k+1}-\varepsilon)dx- \int_\Omega Cudx\\ \ge \frac{1}{2}\|u\|^2 - \frac{\mu_{k+1}-\varepsilon}{2\mu_{k+1}}\|u\|^2- C|\Omega|^{\frac{1}{2}}\left(\int_\Omega |u|^2dx\right)^{\frac{1}{2}} \\ \ge \frac{1}{2}\|u\|^2 \left(1- \frac{\mu_{k+1}-\varepsilon}{\mu_{k+1}}\right)-C|\Omega|^{\frac{1}{2}}\beta_k \|u\|.$

(2.22)

Choosing $r_k:=1/\beta_k$, we easily $r_k\to \infty$ as $k\to \infty$, then

$J(u)\ge \frac{1}{2}\left(1- \frac{\mu_{k+1}-\varepsilon}{\mu_{k+1}}\right) r_k^2-C|\Omega|^{\frac{1}{2}}\to \infty, \text{ as } k\to \infty.$

Hence, $b_k:=\inf\limits_{u\in Z_k, \|u\|=r_k}J(u)\to \infty$ as $k\to \infty$. Combining this and (2.19), we can take $\rho_k:=\max\{d_k,r_k+1\}$, and thus $a_k:=\max\limits_{u\in Y_k, \|u\|=\rho_k}J(u)\le 0$.

Up until now, we have proved the functional $J$ satisfies all the conditions of Lemma 2.2, then $J$ has an unbounded sequence of critical values. Equivalently, (1.1) has infinitely many weak solutions. This completes the proof.