High energy solutions of modified quasilinear fourth-order elliptic equation

This paper focuses on the following modified quasilinear fourth-order elliptic equation:

where \(\triangle^{2}=\triangle(\triangle)\) is the biharmonic operator, \(a>0\), \(b\geq 0\), \(\lambda\geq 1\) is a parameter, \(V\in C(\mathbb{R}^{3},\mathbb{R})\), \(f(x,u)\in C(\mathbb{R}^{3}\times\mathbb{R}, \mathbb{R})\). \(V(x)\) and \(f(x,u)u\) are both allowed to be sign-changing. Under the weaker assumption \(\lim_{ \vert t \vert \rightarrow\infty}\frac{\int^{t}_{0}f(x,s)\,ds}{ \vert t \vert ^{3}}=\infty\) uniformly in \(x\in\mathbb{R}^{3}\), a sequence of high energy weak solutions for the above problem are obtained.

In this paper, we consider the following elliptic equation:

where \(\triangle^{2}=\triangle(\triangle)\) is the biharmonic operator, the constants \(a>0\), \(b\geq 0\), and \(\lambda\geq 1\) is a parameter. \(V(x):\mathbb{R}^{3}\rightarrow \mathbb{R}\) and \(f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}\) satisfies the following assumptions:

\((V)\) :

\(V\in C(\mathbb{R}^{3},\mathbb{R})\), \(\inf_{\mathbb{R}^{3}}V>-\infty\) and there exists a constant \(r>0\) such that

$$\lim_{ \vert y \vert \rightarrow\infty}\operatorname{meas}\bigl\{ x\in\mathbb{R}^{3}: \vert x-y \vert \leq r, V(x)\leq M\bigr\} =0,\quad \forall M>0; $$

\((F_{1})\) :

\(f\in C(\mathbb{R}^{3}\times\mathbb{R},\mathbb{R})\) and there exists positive constant \(C_{0}\) and \(p>4\) such that

$$\bigl\vert f(x,t) \bigr\vert \leq C_{0}\bigl( \vert t \vert + \vert t \vert ^{p-1}\bigr),\quad \forall(x,t)\in \mathbb{R}^{3}\times\mathbb{R}. $$

\((F_{2})\) :

\(\lim_{ \vert t \vert \rightarrow\infty}\frac{F(x,t)}{ \vert t \vert ^{3}}=\infty\) uniformly in \(x\in\mathbb{R}^{3}\), where \(F(x,t)=\int^{t}_{0}f(x,s)\,ds\).

\((F_{3})\) :

There exists a constant \(\alpha\geq 0\) such that

$$f(x,t)t-4F(x,t)\geq-\alpha t^{2}, \quad \forall(x,t)\in \mathbb{R}^{3}\times\mathbb{R}. $$

\((F_{4})\) :

\(f(x,-t)=-f(x,t)\) for all \((x,t)\in\mathbb{R}^{3}\times\mathbb{R}\).

The Kirchhoff’s model considers the changes in length of the string produced by transverse vibrations. It was pointed out in [1–4] that (1.1) models several physical and biological systems where u describes a process which relies on the mean of itself such as the population density. For more mathematical and physical background on Kirchhoff-type problems, we refer the reader to [1, 5–8] and the references therein. It is well known that fourth-order elliptic equation has been widely studied since Lazer and Mckenna [9] first proposed to study periodic oscillations and traveling waves in a suspension bridge.

In te recent years, many scholars widely studied the Schrödinger equation under variant assumptions on \(V(x)\) and \(f(x,u)\), such as [3, 4, 10–13]. In [10], Wu considered the following Schrödinger–Kirchhoff-type problem:

under these hypotheses:

\((V')\) :

\(V\in C(\mathbb{R}^{N},\mathbb{R})\) satisfies \(\inf V(x)\geq a_{1}>0\) and for each \(M>0\), \(\operatorname{meas}\{x\in\mathbb{R}^{N}:V(x)\leq M\}<+\infty\), where \(a_{1}\) is a constant and meas denotes the Lebesgue measure in \(\mathbb{R}^{N}\).

\((f_{1})\) :

\(f\in C(\mathbb{R}^{N}\times\mathbb{R},\mathbb{R})\) and \(\vert f(x,t) \vert \leq C(1+ \vert t \vert ^{p-1})\) for some \(2\leq p<2^{\ast}\), where C is a positive constant;

\((f_{2})\) :

\(f(x,t)=o( \vert t \vert )\) as \(\vert t \vert \rightarrow 0\);

\((f_{3})\) :

\(\frac{F(x,t)}{t^{4}}\rightarrow+\infty\) as \(\vert t \vert \rightarrow+\infty\) uniformly in \(\forall x\in\mathbb{R}^{N}\);

\((f_{4})\) :

\(tf(x,t)\geq4F(x,t)\), \(\forall x\in\mathbb{R}^{N}\), \(\forall t\in\mathbb{R}\).

Here \((f_{3})\) is essential in these references to overcome the missing of compactness. The author got a nontrivial solution of (1.2). In [8], Zhang and Tang also considered the problem (1.2) under the assumption \((V)\), and they obtained infinitely many high energy solutions of the problem (1.2). In [11], Nie studied the following Schrödinger–Kirchhoff-type equation:

under the assumption \((V')\). They got a sequence of high energy weak solutions whenever \(\lambda>0\) is sufficiently large. In [14], Xu and Chen also used condition \((V')\) to study the problem (1.3).

More recently, Cheng and Tang [15] studied the following elliptic equation:

under the assumption \((f_{3})\). Clearly, the problem (1.1) is equivalent to (1.4) whenever \(N=3\), \(a=1\), \(b=0\), \(\lambda=1\), and condition \((f_{3})\) is stronger than \((F_{2})\).

Motivated by the work we discussed above, we will use weaker conditions \((F_{2})\), \((F_{3})\) instead of the common assumptions \((f_{3})\), \((f_{4})\), while \(V(x)\) and \(f(x,u)u\) are both allowed to be sign-changing. We will further study and establish the existence of infinitely many high energy solutions of (1.1) whenever \(\lambda\geq 1\), by using the fountain theorem [16, 17] or its other versions [18, 19]. To the best of our knowledge, there is little work concerning this case up to now.

The following are our main results.

Theorem 1.1

Assume that \((V)\) and \((F_{1})\)–\((F_{4})\) are satisfied, then problem (1.1) possesses infinitely many high energy solutions whenever \(\lambda\geq 1\).

Corollary 1.2

Assume that \((V)\) and \((F_{1})\)–\((F_{4})\) are satisfied, then problem (1.3) possesses infinitely many high energy nontrivial solutions whenever \(\lambda\geq 1\).

Remark 1.3

Obviously, the condition \((V)\) is weaker than \((V')\); \((F_{1})\) is weaker than \((f_{1})\) and \((f_{2})\); \((F_{3})\) is weaker than \((f_{7})\) [14] and \((f_{4})\); \((F_{2})\) is weaker than \((g_{2})\) [15]. Furthermore, we do not require λ large enough, but we only need \(\lambda\geq 1\). Therefore, our results extend and improve Theorem 1 [10], Theorem 1.2 [11], Theorem 1.3 [14], Theorem 1.1 [8], Theorem 1.4 [15] and so on.

Remark 1.4

There are many functions satisfying assumptions \((F_{1})\)–\((F_{4})\) not \((f_{3})\). For example

for all \((x,u)\in\mathbb{R}^{3}\times\mathbb{R}\).

Indeed, \(F(x,u)=u^{4}-\frac{u^{2}\ln(1+u^{2})}{1+u^{2}}\), then we can find a positive constant α such that

In order to apply the variational method, we first recall some related preliminaries and establish a corresponding variational framework for our problem (1.1); then we give the proof of Theorem 1.1.

For \(1< s<+\infty\), define the Sobolev space

equipped with the norm

where \(\alpha=(\alpha_{1}, \alpha_{2},\ldots, \alpha_{N})\) with \(\alpha_{i}\in \mathbb{Z}^{+} \) (the set of all non-negative integers), \(i=1, 2, \ldots, N\), \(\vert \alpha \vert =\alpha_{1}+\alpha_{2}+\cdots+\alpha_{N}\) and

For \(s=2\), \(H^{m}(\mathbb{R}^{N})=W^{m,2}(\mathbb{R}^{N})\) is a Hilbert space equipped with the scalar product

and the norm

Moreover, for \(m=2\) one has

whenever \(u,v \in H^{2}(\mathbb{R}^{N})\).

Under assumption \((V)\), we can find \(V_{0}\geq 0\) such that \(\widetilde{V}(x)=V(x)+V_{0}\geq 1\) for all \(x\in\mathbb{R}^{3}\). Then

is a Hilbert space endowed with the norm

Let

By condition \((V)\), \((F_{1})\) and the fact \(\int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx<\infty\) (see Lemma 2.2 in [20]), \(\Phi_{\lambda}\) is a well-defined class \(C^{1}\) functional. For all \(u,v\in E_{\lambda}\)

Clearly, seeking a weak solution of problem (1.1) is equivalent to finding a critical point of the functional \(\Phi_{\lambda}\).

Definition 2.1

A sequence \(\{u_{n}\}\subset E_{\lambda}\) is said to be a \((C)_{c}\) sequence if

\(\Phi_{\lambda}\) is said to satisfy the \((C)_{c}\) condition if any \((C)_{c}\) sequence possesses a convergent subsequence.

Let \(E'_{\lambda}= \{u\in H^{2}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}(a \vert \nabla u \vert ^{2}+\lambda\widetilde{V}(x)u^{2})\,dx<\infty \}\).

Lemma 2.2

Under assumption \((V)\), the embedding \(E'_{\lambda}\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact for \(2\leq s<2_{\ast}\), where \(2_{\ast}=\frac{2N}{N-4}\), if \(N>4\); \(2_{\ast}=+\infty\), if \(N\leq4\).

Proof

Define

By Propositions 3.1 and 3.3 in [13], we know that the embedding \(E\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact for \(2\leq s<2_{\ast}\) due to the condition \((V)\), and the embedding \(E'_{\lambda}\hookrightarrow E\) is continuous, therefore, the embedding \(E'_{\lambda}\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact for \(2\leq s<2_{\ast}\). □

Lemma 2.3

Under assumptions \((V)\), \((F_{1})\), any bounded \((C)_{c}\) sequence of \(\Phi_{\lambda}\) has a strongly convergent subsequence in \(E_{\lambda}\).

Proof

Let \(\{u_{n}\}\subset E_{\lambda}\) hold with

Then up to a subsequence, there exists a constant \(c\in\mathbb{R}\) such that

According to Lemma 2.2, going if necessary to a subsequence, we can assume that there exists \(u\in E_{\lambda}\) such that

By an elementary computation,

Clearly, \(\lambda V_{0}\int_{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{2}\,dx\rightarrow 0\), and \(\langle\Phi'_{\lambda}(u_{n})-\Phi'(u),u_{n}-u\rangle\rightarrow 0\). Then, since \(\{u_{n}\}\subset E_{\lambda}\) is bounded, we have

Note that \(E_{\lambda}\hookrightarrow H^{2}(\mathbb{R}^{3})\hookrightarrow W^{1,s}(\mathbb{R}^{3})\) for \(2\leq s \leq+\infty\),

where

Applying (2.3)–(2.5) and (2.8), there exist constants \(C_{1}>0\) such that

and \(C'_{1}>0\) such that

By \((F_{1})\) and the Hölder inequality,

Then, combining the last inequality with (2.5), we get

Hence, the combination of (2.7) and (2.9)–(2.11) implies that

Therefore, the proof is complete. □

Lemma 2.4

Assume that \((V)\) and \((F_{1})\)–\((F_{3})\) hold, then \(\Phi_{\lambda}\) satisfies the \((C)_{c}\) condition.

Proof

Let \(\{u_{n}\}\subset E_{\lambda}\) be such that

First, we prove that \(\{u_{n}\}\) is bounded in \(E_{\lambda}\). By \((F_{3})\), (2.1), (2.2) and (2.12), one has

Thus, it remains to show that \(\{u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{3})\). Otherwise, suppose that \(\Vert u_{n} \Vert _{2}\rightarrow\infty\) and then \(\Vert u_{n} \Vert _{\lambda}\rightarrow\infty\). Let \(\omega_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert _{\lambda}}\), then \(\Vert \omega_{n} \Vert _{\lambda}=1\). According to Lemma 2.2, up to a subsequence, for some \(\omega\in E_{\lambda}\), we obtain

Clearly, we deduce that \(\omega\neq 0\) from (2.13). Then, for \(x\in\{y\in\mathbb{R}^{3}:\omega(y)\neq 0\}\), we have \(\vert u_{n}(x) \vert \rightarrow\infty\) as \(n\rightarrow\infty\). For any given \(u\in H^{2}(\mathbb{R}^{3})\backslash\{0\}\), define

By an elementary computation, there exists a unique \(T=\widetilde{t}(u)>0\) such that

This implies that \(g(t)=0\) defines a functional \(T=\widetilde{t}(u)\) on \(H^{2}(\mathbb{R}^{3})\backslash\{0\}\). We define \(\widetilde{t}(0)=0\). It is easy to verify that \(T=\widetilde{t}(u)\) is continuous and \(\widetilde{t}(u)\rightarrow\infty\) as \(\Vert u \Vert _{H^{2}}\rightarrow\infty\).

Due to the definition of g, for any \(u\in H^{2}(\mathbb{R}^{3})\backslash\{0\}\), there exists

such that

Note that \(u_{n}\neq 0\) for large \(n\in\mathbb{N}\), then there exist

such that

That is,

with \(\Vert v_{n} \Vert _{H^{2}}=1\) for large \(n\in\mathbb{N}\). Moreover, we have

and

From \((F_{1})\)–\((F_{3})\), there are \(R_{0}>0\) and \(C_{2}>0\) such that, for all \(x\in\mathbb{R}^{3}\),

and

Thus, by \((F_{3})\), (2.1), (2.2), (2.12)–(2.15) and \(\Vert v_{n} \Vert _{H^{2}}=1\),

By the Hölder inequality and the Sobolev embedding inequality, we see that the sequence of integrals \(\int_{\mathbb{R}^{3}}v_{n}^{2} \vert \nabla v_{n} \vert ^{2}\,dx<\infty\), since \(\Vert v_{n} \Vert _{H^{2}}=1\); on the other hand, by \((F_{2})\) and (2.14), we have

which contradicts (2.16). Hence, \(\{u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{3})\). This shows that \(\{u_{n}\}\) is bounded in \(E_{\lambda}\) due to (2.13). By Lemma 2.3, \(\{u_{n}\}\) contains a convergent subsequence. □

Next, we define

where \(\{e_{j}\}\) is an orthonormal basis of \(E_{\lambda}\).

Lemma 2.5

Assume that \((V)\) holds, then, for \(2\leq s<2_{\ast}\),

Proof

By virtue of Lemma 2.2, we can prove the conclusion in a similar way to [16, Lemma 3.8] and [17, Corollary 8.18]. □

Lemma 2.6

Assume that \((V)\) and \((F_{1})\) hold, then there exist constants \(\rho, \alpha>0\) such that \(\Phi|_{\partial B_{\rho}\cap Z_{m}}\geq\alpha\).

Proof

From (2.1) and \((F_{1})\), for all \(u\in E_{\lambda}\) we have

By virtue of Lemma 2.5, we can choose an integer \(m\geq 1\), for all \(u\in Z_{m}\), satisfying

Combining this with (2.17), one has

Note that, if we let \(\rho= \Vert u \Vert _{\lambda}>0\) be sufficiently small, then \(\Phi_{\lambda}(u)\geq\frac{1}{8}\rho^{2}>0\). □

Lemma 2.7

Assume that \((V)\), \((F_{1})\) and \((F_{2})\) hold, then, for any finite dimensional subspace \(E\subset E_{\lambda}\), there exists \(R=R(E)>0\) such that \(\Phi_{\lambda}|_{E\backslash B_{\rho}}<0\).

Proof

According to the proof of Lemma 2.4, we know that, for any \(u\in E\backslash \{0\}\), there exists a unique \(T=\widetilde{t}(u)>0\) such that

Hence

By the equivalence of norms in the finite dimensional space E, there exists \(C_{3}>0\) such that

Combining this with

we find that, for any \(\delta>0\), there exists a large \(R=R(E,\delta)>0\) such that

By \((F_{1})\), there exists \(C_{4}>0\), for all \(x\in\mathbb{R}^{N}\), \(\vert u \vert \leq R_{0}\) such that

where \(R_{0}\) is given by (2.15). Combining (2.1) with \(\Vert v \Vert _{H^{2}}=1\), it follows that for all \(u\in E\backslash\{0\}\)

Note that \(v\neq0\), then it follows from \((F_{2})\) that

Thus

Combining this with (2.18), we obtain

Thus, there exists a large \(T_{0}>0\) such that

for all \(T\geq T_{0}\). Taking \(\delta=T_{0}\), then there exists a large \(R=R(E)>0\) such that

for all \(u\in E\) with \(\Vert u \Vert _{\lambda}\geq R\).

Hence, \(\Phi_{\lambda}(u)<0\) for all \(u\in E\) with \(\Vert u \Vert _{\lambda}\geq R\). □

Proof of Theorem 1.1

Let \(X=E_{\lambda}\), \(Y=Y_{m}\) and \(Z=\overline{Z_{m}}\). Clearly, \(\Phi(0)=0\) and \(\Phi(u)=\Phi(-u)\) due to \((F_{4})\). By virtue of Lemma 2.4, Lemma 2.6, Lemma 2.7 and the fountain theorem (Theorem 3.6 [16]), problem (1.1) possesses infinitely many high energy solutions. □

Proof of Corollary 1.2

Let us consider the Hilbert space

endowed with the norm

Let

Obviously, Φ is a well-defined class \(C^{1}\) functional, and the embedding \(H\hookrightarrow L^{s}\) is compact for \(2\leq s<6\) (see the proof of Lemma 2.2). By Lemma 2.4, Lemma 2.6, Lemma 2.7 and the fountain theorem (Theorem 3.6 [16]), problem (1.3) possesses infinitely many high energy solutions. □

Remark 2.8

In the next paper, we wish to consider the sign-changing solutions for the biharmonic problem like in [19, 21] and so on.

In this paper, we consider a sequence of high energy weak solutions for the modified quasilinear fourth-order elliptic equation (1.1) under rather weak conditions. We first prove that the energy functional satisfies the Cerami condition in the well-defined Hilbert space and then prove that the fountain theorem holds under the given conditions by a new technique. Our results extend and improve some recent results.

This research was supported by the National Science Foundation of China grant 11471187 and 11571197.

Affiliations

1. School of Mathematical Sciences, Qufu Normal University, Shandong, P.R. China
   • Xiujuan Wang
   • , Anmin Mao
   •  & Aixia Qian

Contributions

All authors contributed equally and significantly in writing this article. All authors wrote, read, and approved the final manuscript.

Corresponding author

Correspondence to Aixia Qian.

Competing interests

The authors declare that they have no competing interests.

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