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High energy solutions of modified quasilinear fourth-order elliptic equation
Boundary Value Problems volume 2018, Article number: 54 (2018)
Abstract
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.
1 Introduction and main results
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
2 Preliminary lemmas and proof of our main result
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.
3 Conclusions
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.
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This research was supported by the National Science Foundation of China grant 11471187 and 11571197.
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Wang, X., Mao, A. & Qian, A. High energy solutions of modified quasilinear fourth-order elliptic equation. Bound Value Probl 2018, 54 (2018). https://doi.org/10.1186/s13661-018-0970-6
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DOI: https://doi.org/10.1186/s13661-018-0970-6