Infinitely many solutions for p-biharmonic equation with general potential and concave-convex nonlinearity in \(\mathbb{R}^{N}\)

In this paper, we study the existence of multiple solutions to a class of p-biharmonic elliptic equations, \(\Delta^{2}_{p}u-\Delta_{p}u+V(x)|u|^{p-2}u =\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u\), \(x\in \mathbb{R}^{N}\), where \(1< m< p< q< p_{*}=\frac {pN}{N-2p}\), \(\Delta^{2}_{p} u=\Delta(|\Delta u|^{p-2}\Delta u)\) is a p-biharmonic operator and \(\Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\). The potential function \(V(x)\in C{({\mathbb{R}}^{N})}\) satisfies \(\inf_{x\in{\mathbb{R}} ^{N}}V(x)>0\). By variational methods, we obtain the existence of infinitely many solutions for a p-biharmonic elliptic equation in \({\mathbb{R}}^{N}\).

In this paper, we are interested in the existence of solutions to the following p-biharmonic elliptic equation:

where \(2<2p<N\), \(f(x,u)=\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u\), \(1< m< p< q< p_{*}=\frac{pN}{N-2p}\), \(\Delta^{2}_{p} u=\Delta(|\Delta u|^{p-2}\Delta u)\) is a p-biharmonic operator and \(\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\). The potential function \(V(x)\in C{({\mathbb{R}}^{N})}\) satisfies \(\inf_{x\in{\mathbb{R}}^{N}}V(x)>0\).

Recently, the nonlinear biharmonic equation in an unbounded domain has been extensively investigated, we refer the reader to [1–9] and the references therein. For the whole space \({\mathbb{R}}^{N}\) case, the main difficulty of this problem is the lack of compactness for the Sobolev embedding theorem. In order to overcome this difficulty, the authors always assumed the potential \(V(x)\) has some special characteristic. For example, in [4], Yin and Wu studied the following fourth-order elliptic equation:

where the potential \(V(x)\) satisfied

(V₀)::

\(V\in C({\mathbb{R}}^{N})\) satisfies \(\inf_{x\in{\mathbb {R}}^{N}}V(x)>0\), and for each \(M>0\), \(\operatorname{meas}\{x\in \mathbb{R}^{N}\leq M\}<+\infty\).

where the potential \(V(x)\) satisfied a weaker condition than (V₀), that is,

(V₁)::

\(V\in C({\mathbb{R}}^{N})\) satisfies \(\inf_{x\in{\mathbb {R}}^{N}}V(x)>0\), there exists some \(M>0\), \(\operatorname{meas}\{x\in \mathbb{R}^{N}\leq M\}<+\infty\).

(V)::

\(V(x)\in C({\mathbb{R}}^{N})\) satisfies \(\inf_{x\in {\mathbb{R}}^{N}}V(x)>0\).

In the present paper, we will answer this interesting question. We consider the existence of solutions to the p-biharmonic problem (1.1) with a more general potential \(V(x)\). To prove that the \((\mathit{PS})\) sequence weakly converges to a critical point of the corresponding functional, we adapt ideas developed by [10–12] and then by variational methods, we establish the existence of infinitely many high-energy solutions to problem (1.1) with a concave-convex nonlinearity, i.e., \(f(x,u)=\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u\), \(1< m< p< q< p_{*}=\frac{pN}{N-2p}\). To the best of our knowledge, little has been done for p-biharmonic problems with this type of nonlinearity. Here, we give our assumptions on the weight functions \(h_{1}(x)\) and \(h_{2}(x)\):

(H₁):

\(h_{1}\in L^{\sigma}({\mathbb{R}}^{N})\) with \(\sigma=\frac {p}{p-m}\);

(H₂):

\(h_{2}(x)\geq0\) (≢0), \(h_{2}(x)\in L^{\infty}({\mathbb{R}}^{N})\).

The main result in this paper is as follows.

Theorem 1.1

Let \(2<2p<N\), \(1< m< p< q< p_{*}=\frac{pN}{N-2p}\). Assume (V), (H₁), and (H₂) hold. Then there exists \(\lambda_{0}>0\) such that for all \(\lambda\in [0,\lambda_{0}]\), problem (1.1) admits infinitely many high-energy solutions in \({\mathbb{R}}^{N}\).

This paper is organized as follows. In Section 2, we build the variational framework for problem (1.1) and establish a series of lemmas, which will be used in the proof of Theorem 1.1. In Section 3, we prove Theorem 1.1 by the mountain pass theorem [13].

In order to apply the variational setting, we assume the solutions of (1.1) belong to the following subspace of \(\mathcal {D}^{2,p}({\mathbb{R}}^{N})\):

endowed with the norm

where \(\mathcal{D}^{2,p}({\mathbb{R}}^{N})=\{u\in L^{p_{*}}({\mathbb {R}}^{N}) |\Delta u\in L^{p}({\mathbb{R}}^{N})\}\), \(\|\cdot\|_{s}\) means the norm in \(L^{s}({\mathbb{R}}^{N})\).

We denote by \(S_{*}\) the Sobolev constant, that is,

and

where \(S_{*}\) is obtained by a positive and radially symmetric function; see for instance [14].

Definition 2.1

A function \(u\in E\) is said to be a weak solution of (1.1) if, for any \(\varphi\in E\), we have

Let \(J(u):E\to{\mathbb{R}}\) be the energy functional associated with problem (1.1) defined by

From the embedding inequality (2.4) and the assumptions in Theorem 1.1, we see the functional \(J\in C^{1}(E,{\mathbb{R}})\) and its Gateaux derivative is given by

To prove the existence of infinitely many solutions to problem (1.1), we need to prove that the functional J defined by (2.6) satisfies the \((\mathit{PS})\) condition. Recall that a sequence \(\{u_{n}\}\) in E is called a \((\mathit{PS})_{c}\) sequence of J if

The functional J satisfies the \((\mathit{PS})\) condition if any \((\mathit{PS})_{c}\) sequence possesses a convergent subsequence in E.

Lemma 2.1

Assume (V), (H₁), and (H₂) hold. If \(\{u_{n}\}\subset E\) is a \((\mathit{PS})_{c}\) sequence of J, then \(\{u_{n}\}\) is bounded in E.

Proof

It follows from Hölder’s inequality that

where \(a_{1}=V_{0}^{-{m}/{p}}\|h_{1}\|_{\sigma}\). Choose \(t\in(0,1)\) such that \(q=pt+(1-t)p_{*}\), then

where \(a_{2}=S_{*}^{-{p_{*}(q-p)}/{p(p_{*}-p)}}V_{0}^{-t}\|h_{2}\|_{\infty}\). Thus,

Since \(1< m< p< q\), we conclude that \(\|u\|_{E}\) is bounded and the proof is complete. □

In the following, we shall show that \(\{u_{n}\}\) has a convergent subsequence in E. Since the sequence \(\{u_{n}\}\) given by (2.8) is a bounded sequence in E, there exist a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)) and \(v\in E\) such that \(\|u_{n}\|_{E}\leq M\), \(\|v\|_{E}\leq M\), and

Lemma 2.2

Assume (V), (H₁), and (H ₂) hold. If the sequence \(\{u_{n}\}\) is bounded in E satisfying (2.12), then

1. (i)

   \(\lim_{n\to\infty}\int_{{\mathbb{R}} ^{N}}h_{1}(x)|u_{n}|^{m}\,dx=\int_{{\mathbb{R}}^{N}}h_{1}(x)|v|^{m}\,dx\), \(\lim_{n\to \infty}\int_{{\mathbb{R}}^{N}}h_{1}(x)|u_{n}-v|^{m}\,dx=0\);

2. (ii)

   \(\lim_{n\to\infty}\int_{{\mathbb{R}} ^{N}}h_{2}(x)|u_{n}|^{q}\,dx=\int_{{\mathbb{R}}^{N}}h_{2}(x)|v|^{q}\,dx\), \(\lim_{n\to \infty}\int_{{\mathbb{R}}^{N}}h_{2}(x)|u_{n}-v|^{q}\,dx=0\).

Proof

(i) In fact, from \(h\in L^{\sigma}({\mathbb{R}}^{N})\) and (2.12), we obtain, for any \(r>0\),

where and in the sequel \(B_{r}=\{x\in{\mathbb{R}}^{N}:|x|< r\}\), \(B^{c}_{r}={\mathbb{R}}^{N}\setminus \overline{B}_{r}\). On the other hand, we see from the Hölder inequality that

as \(r\to\infty\). By Fatou’s lemma, we see that, as \(n\to\infty\),

Then, the application of (2.13)-(2.15) gives the first limit of (i). Furthermore, by the Brezis-Lieb lemma in [15], we have the second limit of (i).

(ii) To prove the conclusion (ii), we follow the argument used in [10–12]. Here, we give a detailed proof for the reader’s convenience.

Since \(p< p< p_{*}\), it easy to see that, for any small \(\varepsilon>0\), there exist \(S_{0}>s_{0}>0\) such that \(|s|^{q}<\varepsilon|s|^{p}\) if \(|s|\leq s_{0}\) and \(|s|^{q}\leq\varepsilon|s|^{p_{*}}\), if \(|s|\geq S_{0}\). This shows that

Denote \(A_{n}=\{x\in{\mathbb{R}}^{N};s_{0}\leq|u_{n}(x)|\leq S_{0}\}\). It follows from (2.16), (2.4), and (2.12) that

with some constant \(M_{1}>0\), and

where \(|A_{n}|=\operatorname{meas}(A_{n})\). Equation (2.18) implies that \(\sup_{n\in{\mathbb{N}}}|A_{n}|\leq M_{1}|s_{0}|^{-p_{*}}<\infty\), so it is easy to see that

In the following, we show that \(\lim_{r\to\infty }\operatorname{meas}(A_{n}\cap B_{r}^{c})=0\) uniformly in \(n\in{\mathbb{N}}\).

In fact, it follows from (2.12) that \(v\in L^{p}({\mathbb{R}}^{N})\) and \(u_{n}(x)\to v(x)\) a.e. \({\mathbb{R}}^{N}\). Therefore, for any small \(\varepsilon >0\), there exists \(r_{0}>1\) such that \(r\geq r_{0}\),

For this ε, we choose \(t_{1}=r_{0}\), \(t_{j}\uparrow\infty\) such that \(D_{j}=B^{c}_{t_{j}}\setminus\overline{B}^{c}_{t_{j+1}}\), \(B^{c}_{r_{0}}=\bigcup^{\infty}_{j=1}D_{j}\) and

Obviously, for every fixed \(j\in N\), \(D_{j}\) is a bounded domain and \(D_{j}\cap D_{i}=\emptyset\) (\(j\neq i\)). Furthermore, \(s_{0}\leq|u_{n}|\leq S_{0}\) in \(D_{j}\cap A_{n}\). By Fatou’s lemma, we have, for every \(j\in {\mathbb{N}}\),

Then, for \(s_{1}=2^{1-q}s_{0}^{q}\), we obtain

Notice that, for any \(r\geq r_{0}\) and \(n\in{\mathbb{N}}\), we have \((A_{n}\cap B^{c}_{r})\subset(A_{n}\cap B_{r_{0}}^{c})\). Therefore, the application of (2.19) and (2.20) yields \(\lim_{r\to\infty}|A_{n}\cap B_{r}^{c}|=0\) uniformly in \(n\in{\mathbb{N}}\). Thus, for any \(\varepsilon>0\), there exists \(r_{0}\geq1\) such that \(\operatorname{meas}(A_{n}\cap B^{c}_{r})<\frac{\varepsilon}{S_{0}^{q}\|h_{2}\| _{\infty}}\), for \(r\geq r_{0}\). Then it follows from (2.17) that

and

Moreover, we derive from (2.12) that

Therefore, using (2.21) and (2.22), and the application of Brezis-Lieb lemma in [15] we conclude the second limit of (ii). Then the proof is complete. □

Lemma 2.3

Let \(\{u_{n}\}\) be a \((\mathit{PS})_{c}\) sequence satisfying (2.12), then \(u_{n}\to v\) in E, that is, the functional J satisfies the \((\mathit{PS})\) condition.

Proof

Denote

Then the fact \(J'(u_{n})\to0\) in \(E^{*}\) shows that \(P_{n}\to0\) as \(n\to \infty\). Moreover, the fact \(u_{n}\rightharpoonup v\) in E implies \(Q_{n}\to0\), where

It follows from the Hölder inequality and the limit (i) in Lemma 2.2 that

Similarly, we can derive from the limit (ii) in Lemma 2.2 that

Then (2.24) and (2.25) show that as \(n\to\infty\)

Then we have \(\|u_{n}-v\|_{E}\to0\) as \(n\to\infty\). Thus \(J(u)\) satisfies the \((\mathit{PS})\) condition on E and the proof is completed. □

In this section, we will give the proof of Theorem 1.1. We assume that all conditions in the theorem hold. The proof mainly relies on the mountain pass theorem.

Lemma 3.1

([13])

Let E be an infinite dimensional real Banach space, \(J\in C^{1}(E,{\mathbb{R}})\) be even and satisfies the \((\mathit{PS})\) condition and \(J(0)=0\). Assume \(E=Y\oplus Z\), Y is finite dimensional, and J satisfies:

(J₁):

There exist constants \(\rho,\alpha>0\) such that \(J(u)\leq \alpha\) on \(\partial B_{\rho}\cap Z\).

(J₂):

For each finite dimensional subspace \(E_{0}\subset E\), there is an \(R_{0}=R_{0}(E_{0})\) such that \(J(u)\leq0\) on \(E_{0}\setminus B_{R_{0}}\), where \(B_{r}=\{u\in E: \|u\|_{E}< r\}\).

Proof of Theorem 1.1

Clearly, the functional J defined by (2.6) is even in E. By Lemma 2.2 in Section 2, the functional satisfies the \((\mathit{PS})\) condition. Next, we prove that J satisfies (J₁) and (J₂). From (2.9) and (2.10), it follows that

Denote

Then, there exist \(\lambda_{0},z_{1},\alpha>0\) such that \(\phi(z_{1})\geq \alpha\) for any \(\lambda\in[0,\lambda_{0}]\). Let \(\rho=z_{1}\), we have \(J(u)\geq\alpha\) with \(\|u\|_{E}=\rho\) and \(\lambda\in[0,\lambda _{0}]\). So the condition (J₁) is satisfied.

We now verify (J₂). For any finite dimensional subspace \(E_{0}\subset E\), we assert that there is a constant \(R_{0}>\rho\) such that \(J<0\) on \(E_{0}\setminus B_{R_{0}}\). Otherwise, there exists a sequence \(\{u_{n}\} \subset E_{0}\) such that \(\|u\|_{n}\to\infty\) and \(J(u_{n})\geq0\). Hence

Set \(\omega_{n}=\frac{u_{n}}{\|u_{n}\|_{E}}\). Then up to a sequence, we can assume \(\omega_{n}\rightharpoonup\omega\) in E, \(\omega_{n}\rightarrow \omega\) a.e. in \({\mathbb{R}}^{N}\). Denote \(\Omega=\{x\in{\mathbb {R}}^{N}:\omega(x)\neq 0\}\). Assume \(|\Omega|>0\). Clearly, \(u_{n}(x)\to\infty\) in Ω. It follows from (2.8) and (2.9) that

On the other hand, we derive

Therefore, multiplying (3.1) by \(\|u\|_{E}^{-p}\) and passing to the limit as \(n\to\infty\) show that \(\frac{1}{p}\geq\infty\). This is impossible. So \(|\Omega|=0\) and \(\omega(x)=0\) a.e. on \({\mathbb {R}}^{N}\). By the equivalence of all norms in \(E_{0}\), there exists a constant \(\beta >0\) such that

Hence

This is a contradiction. So there exists a constant \(R_{0}\) such that \(J<0\) on \(E_{0}\setminus B_{R_{0}}\). Therefore, the existence of infinitely many solutions \(\{u_{n}\}\) for problem (1.1) follows from Lemma 3.1 and we finish the proof of Theorem 1.1. □

The authors wish to express their gratitude to the referees for valuable comments and suggestions. This work was supported by the Project of Innovation in Scientific Research for Graduate Students of Jiangsu Province (No. CXZZ13-0263) and the Fundamental Research Funds for the Central Universities of China (2015B31014).

Affiliations

1. College of Science, Hohai University, Focheng Road, Nanjing, 210098, P.R. China
   • Lihua Liu
   •  & Caisheng Chen

Corresponding author

Correspondence to Lihua Liu.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors contributed to each part of this study equally. All authors read and approved the final vision of the manuscript.

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