Infinitely many solutions for a class of biharmonic equations with indefinite potentials

1.   Introduction and main result

In this paper, we consider the following sublinear biharmonic equations

where $N\geq5, \; 0 < p < 1$, $V(x), K(x)\in L^{\infty}({\mathbb R}^{N})$ both change sign in ${\mathbb R}^{N}$ and satisfies some conditions specified below. Problem (1.1) involved biharmonic operator arises in the study of traveling waves in suspension bridge. Furthermore, it is well know that biharmonic operator arises in the study of static deflection of a plate, for more details, we refer the read to ^[1,2,3].

For biharmonic equations, there have been many results ^{[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]} and the references therein. In these results, some authors studied biharmonic equations on the whole space ${\mathbb R}^{N}$ ^{[4,6,9,10,12,14,15,16,17,19,20,21]}, in which most of them were focused on superlinear case but few results involving sublinear case ^{[10,15,16,20,21]}. On the other hand, we notice that, in ^[4,19], authors considered the biharmonic equations under sign-changing potential and superlinear case. However, to our best knowledge, there are no results of biharmonic equations on ${\mathbb R}^{N}$ in case of sign-changing potential and sublinear case. In this paper, we will investigate the nontrivial solutions for Eq 1.1 with the potential indefinite in sign and sublinear case, the tool used in our paper is the symmetric Mountain Pass Theorem.

To stated our main result, we assume that:

$(H_{1})$ $V\in L^{\infty}({\mathbb R}^{N})$ and there exist $\alpha, R_{0} > 0$ such that

$(H_{2})$ $\|V^{-}\|_{\frac{N}{4}} < \frac{1}{S}$, where $V^{\pm}(x) = \max\{\pm V(x), 0\}$ and the $S$ is the constant of Sobolev:

$(H_3)$ $K\in L^{\infty}({\mathbb R}^{N})$ and there exist $\beta > 0, R_{1} > R_{2} > 0$, $y_{0} = (y_{1}, ..., y_{N})\in {\mathbb R}^{N}$ such that

Our main result is as follows:

Theorem 1.1. Assume $(H_{1})-(H_{3})$ hold. Then problem $(1.1)$ possesses infinitely many nontrivial solutions.

For biharmonic equations on the whole space ${\mathbb R}^{N}$, the main difficulty one may face is the Sobolev embedding $H^{2}({\mathbb R}^{N})\hookrightarrow L^{s}({\mathbb R})$ is not compact for $s\in [2, 2_{\ast})$. To overcome this difficulty, one can restrict the corresponding energy functional to a subspace of $H^{2}({\mathbb R}^{N})$, which embeds compactly into $L^{s}({\mathbb R}^{N})$ with certain qualifications or consisting of radially symmetric functions. For example, Yin and Wu ^[17] and Ye and Tang ^[15] considered biharmonic equations with the potential $V$ satisfying following conditions:

$(V_{1})$ $V\in C({\mathbb R}^{N}, {\mathbb R}), \; \inf_{x\in {\mathbb R}^{N}}V(x)\geq b > 0\; \text{and for each}\; M > 0, \; meas\{x\in {\mathbb R}^{N}: V(x)\leq M\} < \infty,$ where $b$ is a constant and $meas$ denotes Lebesgue measure in ${\mathbb R}^{N}$.

In fact, due to the condition $(V_{1})$, space

can embed compactly into $L^{s}({\mathbb R}^{N})$ for $s\in [2, 2_{\ast}),$ which is crucial in their paper.

Subsequently, Liu, Chen and Wu ^[6] and Ye and Tang ^[16] studied biharmonic equations with $\lambda V$ instead of potential $V$ under more weaker condition than $(V_{1})$, i.e.,

$(V_{2})$ $V\in C({\mathbb R}^{N}, {\mathbb R}), \; \inf_{x\in {\mathbb R}^{N}}V(x)\geq b > 0\; \text{there exists}\; M > 0, \; meas\{x\in {\mathbb R}^{N}: V(x)\leq M\} < \infty,$ where $b$ is a constant and $meas$ denotes Lebesgue measure in ${\mathbb R}^{N}$.

Under the condition $(V_{2})$, it is obvious that $V(x)$ no longer satisfies certain coercive condition. Hence, corresponding Sobolev's embedding is not compact. Fortunately, with the aid of parameter $\lambda$ ($\lambda > 0$ large enough), they obtained that the corresponding energy functional possess the property of locally compact.

In ^[20,21], Zhang, Tang and Zhang considered biharmonic equations on ${\mathbb R}^{N}$ under more weaken conditions than $(V_{1})$ and $(V_{2})$. However, their results does not allow $V(x)$ to change sign.

Recently, Su and Chen ^[10] studied the following sublinear biharmonic equation

where $N > 4, \lambda > 0, 1 < q < 2$ and $\mu\in[0, \mu_{0}]$. By using Ekeland's variational principle and Gigliardo-Nirenberg's inequality, they proved the existence of nontrivial solution for the above problem. It is noticed that their results require conditions $V(x)\geq0$ and $K(x) > 0$.

However, in this paper, the condition like type $(V_{1})$ or $(V_{2})$ does not be needed. Furthermore, functions $V(x)$ and $K(x)$ both change sign in ${\mathbb R}^N$. So, the conditions in this paper more weaken than that of in ^{[6,10,15,16,17,20,21]}. On the other hand, our main result also supplement the results obtained by ^[4,19] in which the sign-changing potential and superlinear case were considered. It is worth pointing out that there are some interesting results, for example ^{[23,24,25,26,27,28,29,30,31,32,33,34,35,36]}, considered elliptic equations with an indefinite nonlinearity or sublinear condition or nonlocal terms.

2.   Notations and preliminaries

In this paper, we use the following notations. Let

Let $E$ be a Banach space and $\varphi:E\rightarrow {\mathbb R}$ be a functional of class $C^{1}$, the Fréchet derivative of $\varphi$ at $u$, $\varphi'(u)$, is an element of the dual space $E^{\ast}$ and we denote $\varphi'(u)$ evaluated at $v\in E$ by $\langle\varphi'(u), v\rangle$.

The Sobolev space $E = H^{2}({\mathbb R}^{N})\cap L^{p+1}({\mathbb R}^{N}), 0 < p < 1,$ endowed with the norm by

Obviously, the space $E$ is a reflexive Banach space.

The energy functional $\varphi:E\rightarrow {\mathbb R}$ corresponding to problem $(1.1)$ is defined by

It is well know that, under our conditions, $\varphi\in C^{1}(E)$ and its critical points are solutions of problem $(1.1)$.

Definition 2.1. (^[37]) Let $E$ be a Banach space and $A$ a subset of $E$. Set $A$ is said to be symmetric if $u\in E$ implies $-u\in E$. For a closed symmetric set $A$ which does not contain the origin, we define a genus $\gamma(A)$ of $A$ by the smallest integer $l$ such that there exist an odd continuous mapping from $A$ to ${\mathbb R}^{l}\setminus \{0\}$. If there does not exists such a $l$, we define $\gamma(A) = \infty.$ We set $\gamma(\emptyset) = 0$. Let $\Gamma_{l}$ denote the family of closed symmetric subsets $A$ of $E$ such that $0\notin A$ and $\gamma(A)\geq l.$

The following result is a version of the classical symmetric Mountain Pass Theorem ^[37,38]. A proof can be found in ^[39].

Theorem 2.1. (^[39]) Let $E$ be an infinite dimensional Banach space and let $\varphi\in C^{1}(E, {\mathbb R})$ satisfy:

$(1)$ $\varphi$ is even, bounded from below, $\varphi(0) = 0$ and $\varphi$ satisfies the Palais-Smale condition.

$(2)$ For each $l\in {\mathbb N}$, there exists an $A_{l}\in \Gamma_{l}$ such that

Then either of the following two conditions holds:

$(i)$ There exists a sequence ${u_{l}}$ such that $\varphi'(u_{l}) = 0, \varphi(u_{l}) < 0$ and ${u_{l}}$ converges to zero; or

$(ii)$ There exist two sequences ${u_{l}}$ and ${v_{l}}$ such that $\varphi'(u_{l}) = 0, \varphi(u_{l}) = 0, u_{l}\neq 0,$ $\lim_{l\rightarrow+\infty}u_{l} = 0, \varphi'(v_{l}) = 0, \varphi(v_{l}) < 0, \lim_{l\rightarrow+\infty}\varphi(v_{l}) = 0$ and ${v_{l}}$ converges to a non-zero limit.

3.   Proof of theorem 1.1

Lemma 3.1. If $(H_{1})-(H_{3})$ hold. Then any PS sequence of $\varphi$ is bounded in $E.$

Proof. Let $\{u_{n}\}\subset E$ be such that

That is, there exists $C > 0$ such that $\varphi(u_{n})\leq C$. Then, according to $(H_3)$, Hölder inequality and Sobolev embedding, one has that

So, thanks to $0 < p < 1$, there exists $\eta > 0$ such that

On the other hand, one has that

By $(H_{3})$, we have that

Then, thanks to $K\in L^{\infty}(\mathbb{R}^N)$, we have that

So, by Hölder inequality and Soblev inequality, we have that

By $(H_{3})$ again, we know that $K^{-}(x)\geq\beta,$ for all $|x| > R_{1}$. Then, we have that

Thanks to (3.1), (3.2) and (3.3), there is a constant $C_{1} > 0$ such that

Since $0 < p < 1$, $\{u_{n}\}$ is bounded in $L^{p+1}({\mathbb R}^{N})$.

Then, from (3.1), we conclude that $\{u_{n}\}$ is bounded in $E$.

Lemma 3.2. (^[40]) Let $x, y$ be for all real numbers, there exists a constant $c > 0$ such that

Lemma 3.3. If $(H_{1})-(H_{3})$ hold, then $\varphi$ satisfies the $PS$ condition on $E$.

Proof. Let $\{u_{n}\}\subset E$ be such that

According to Lemma 3.1, $\{u_{n}\}$ is bounded in $E$. Passing to a subsequence in necessary, we can assume that

So, for any $h\in C^{\infty}_{0}({\mathbb R}^{N}),$ one has that

On the other hand, by Sobolev embedding and Lebesgue's dominated convergence theorem, one has that

Therefore, according to above facts, one has that

Hence, we have that

Let $v_{n} = u_{n}-u,$ then $u_{n} = v_{n}+u$, and we have that

By Lemma 3.2, we have that

We claim that

as $n\rightarrow+\infty.$

In fact, because $E\hookrightarrow L^{p+1}(\mathbb{R}^N)$ is continuous and $v_{n}\rightharpoonup 0$ in $E$, we obtain that $v_{n}\rightharpoonup 0$ in $L^{p+1}(\mathbb{R}^N)$. On the other hand, it is obvious that $|u|^{p}\in L^{\frac{p+1}{p}}(\mathbb{R}^N)$. So, by $K\in L^{\infty}(\mathbb{R}^N)$ and definition of weakly convergence in space $L^{p+1}(\mathbb{R}^N)$, we have that

as $n\rightarrow+\infty.$

Hence, according to (3.4) and (3.5), we have that

Then, we obtain that

Claim 1: $\int_{\mathbb{R}^N}V^{-}(x)v_{n}^{2}dx\rightarrow 0$ as $n\rightarrow+\infty.$

In fact, by $(H_{1})$, we have that $V^{-}(x) = 0$, for all $|x|\geq R_{0}$. So, from $v_{n}\rightarrow 0 \; \text{in}\; L^{q}_{loc}(\mathbb{R}^N), 2\leq q < 2_{\ast},$ and $V\in L^{\infty}(\mathbb{R}^N)$, we obtain $\int_{\mathbb{R}^N}V^{-}(x)v_{n}^{2}dx\rightarrow 0$ as $n\rightarrow+\infty.$

Claim 2: $\int_{\mathbb{R}^N}\left(K^{+}(x)+\chi_{B(0, R_{1})}(x)\right)|v_{n}|^{p+1}dx\rightarrow 0$ as $n\rightarrow+\infty.$

In fact, by $(H_{3})$, we have that $K^{+}(x) = 0$, for all $|x| > R_{1}$. Thanks to $K\in L^{\infty}(\mathbb{R}^N)$ and $v_{n}\rightarrow 0 \; \text{in}\; L^{q}_{loc}(\mathbb{R}^N), 2\leq q < 2_{\ast},$ we get

as $n\rightarrow+\infty.$

Combining claim 1, claim 2, (3.3) and (3.6), we obtain that

That is, $v_{n}\rightarrow0$ in $E$. The proof is complete.

The proof of following Lemma is based on some ideas of Kajikiya ^[39] and very similar to the one contained in ^[40]. For reader convenient, we give the proof.

Lemma 3.4. If $(H_{1})-(H_{3})$ hold, then for each $l\in {\mathbb N}$, there exists subset $A_{l}\in\Gamma_{l}$ such that,

Proof. For $R_{2}$ and $y_{0}$ given by $(H_{3})$, let

Let $l\in {\mathbb N}$ be an arbitrary number and define $n = \min\{n\in {\mathbb N}:n^{N}\geq l\}$. By planes parallel to each face of $B(R_{2})$, $B(R_{2})$ be equally divided into $n^{N}$ small partes $B_{i}$ with $1\leq i\leq n^{N}$. In fact, the length $a$ of the edge $B_{i}$ is $\frac{R_{2}}{n}$. Let $F_{i}\subset B_{i}$ be new cubes such that $F_{i}$ has the same center as that of $B_{i}$. The faces of $F_{i}$ and $B_{i}$ are parallel, and the length of the edge of $F_{i}$ is $\frac{a}{2}$. Let $\phi_{i}\in C({\mathbb R}^{N})$, $1\leq i \leq l$, satisfy: $supp(\phi_{i})\subset B_{i}$; $supp(\phi_{i})\cap supp(\phi_{j}) = \emptyset\; (i\neq j)$; $\phi_{i}(x) = 1$ for $x\in F_{i}$; $0\leq \phi_{i}(x)\leq 1$, for all $x\in \mathbb{R}^{N}$. Let

According to the fact that the mapping $(t_{1}, \cdots, t_{l})\rightarrow \sum_{i = 1}^{l}t_{i}\phi_{i}$ from $S^{l-1}$ to $W_{l}$ is odd and homeomorphic, so $\gamma(W_{l}) = \gamma(S^{l-1}) = l$. Since $W_{l}$ is compact in $E$, it follows that there exists $\alpha_{l} > 0$ such that

On the other hand, we claim that

where $s = \frac{2_{*}(1-p)}{2(2_{*}-p-1)}$.

First, we prove that $\|u\|_{2_{\ast}}\leq C\|\Delta u\|_{2}, \forall u \in E.$

In fact, $\forall u\in E$, $u\in H^{2}(\mathbb{R}^{N})$, so $u, \nabla u\in H^{1}(\mathbb{R}^{N})$. Then, by Gagliardo-Nirenberg-Sobolev inequality (since $C_{0}^{\infty}(\mathbb{R}^{N})$ is dense in $H^{1}(\mathbb{R}^{N})$), we have

where $2^{\ast} = \frac{2N}{N-2}$ and $C_{1}$ depending only on $N$.

Since $2 < 2^{\ast} < N$, by using Gagliardo-Nirenberg-Sobolev inequality again, we have

where $C_{2}$ depending on $N$.

Next, since $1 < p+1 < 2 < 2_{\ast}$, by interpolation inequality, we have

where $\frac{1}{2} = \frac{s}{2_{\ast}}+\frac{1-s}{p+1}$ (thai is $s = \frac{2_{*}(1-p)}{2(2_{*}-p-1)}$).

So $\|u\|_{2}\leq C\|\Delta u\|^{s}_{2}\|u\|^{1-s}_{p+1}\leq C\|u\|$.

According to above facts, there exists $c_{l} > 0$ such that

Let $t > 0$ and $v = \sum_{ = 1}^{l}t_{i}\phi_{i}(x)\in W_{l},$

From (3.2), there exists $j\in[1, l]$ such that $|t_{j}| = 1$ and $|t_{i}|\leq1$ for $i\neq j$. So

According to $\phi_{j}(x) = 1$ for $x\in F_{j}$ and $|t_{j}| = 1$, one has that

By $(H_{3})$, one has that

So, combining (3.3), (3.4), (3.5) and (3.6), we have that

Therefore, it is easy to see that

Hence, we can fixed $t$ small enough such that $\sup\{\varphi(v), v\in A_{l}\} < 0,$ where $A_{l} = tW_{l}\in\Gamma_{l}$.

Lemma 3.5. If $(H_{1})-(H_{3})$ hold. Then $\varphi$ is bounded from below.

Proof. By $(H_3)$, Hölder inequality and Sobolev embedding, as in the proof of Lemma 3.1, we have that

Since $0 < p < 1$, we conclude the proof.

The proof of Theorem 1.1

Proof. In fact, $\varphi(0) = 0$ and $\varphi$ is an even functional. Then by Lemma 3.3, Lemma 3.4 and Lemma 3.5, the conditions $(1)$ and $(2)$ of Theorem 2.1 are satisfied. Therefore, by Theorem 2.1, problem $(1.1)$ possesses infinitely many nontrivial solutions converging to $0$ with negative energy.

Remark 3.1. By using Theorem 2.1, we obtain infinitely many nontrivial solutions to problem $(1.1)$. For infinitely many nontrivial solutions converges to $0$, we must verify that the functional $\varphi$ satisfies some assumption like (A3) in ^[39], (see Remark 1.2 in ^[39]). In fact, we can verify this property by following similar inequality obtained in [^[40], page 460].

4.   Conclusions

In this paper, by using the symmetric Mountain Pass Theorem, we prove a class of biharmonic equations with indefinite potentials has infinitely many solutions. Because our result mainly involves theoretical research, we don't know how to use our result to the real applications in practical problems. So, we should pay attention to both theoretical research and practical application in the follow-up work.

Acknowledgments

The paper is supported by the Natural Science Foundation of China (Grant no. 11561043, 11961043) and Natural Science Foundation of China (Grant no. 11501318).

Conflict of interest

The authors declare no conflict of interest in this paper.