Abstract
This paper is concerned with the following nonlocal fourth-order elliptic problem:
by using the mountain pass theorem, the least action principle, and the Ekeland variational principle, the existence and multiplicity results are obtained.
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1 Introduction
In this paper, we consider the following nonlocal fourth-order elliptic problem:
where \(\varOmega \subset R^{N}\) (\(N>4\)) is a bounded smooth domain, \(m(\cdot )\in C(R^{+},R^{+})\), \(a(\cdot )\in C(\overline{\varOmega },R^{+})\), \(s\in (1,2)\), and \(f\in C(\overline{\varOmega }\times R,R)\).
Problem (1.1) is related to the stationary problems associated with
This plate model was proposed by Berger [1] in 1955, as a simplification of the von Karman plate equation which describes large defection of a plate, where the parameter Q describes in-plane forces applied to the plate and the function f represents transverse loads which may depend on the displacement u and the velocity \(u_{t}\).
Because of the important background, several researchers have considered problem (1.1) by using variational methods when \(a(x)\equiv 0\),
with the function m being bounded or unbounded and f having superlinear growth. We refer the readers to [2–11] and the references therein.
Recently, in [12], Ru et al. considered problem (1.1) with \(m(t)=a+bt\) and a more general f such as
By using an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al. [13], they proved that the above problem has at least one nontrivial solution.
One of the important conditions in their work is that \(f(x,t)\) satisfies the famous Ambrosetti–Rabinowitz type condition, for short, which is called the (AR) condition:
- (AR condition):
-
there exist \(\varTheta >2\) and \(t_{1}>0\), such that
$$ 0< \varTheta F(x,t,\xi _{1},\xi _{2})\leq tf(x,t,\xi _{1},\xi _{2}), \quad \forall \vert t \vert \geq t_{1},x\in \varOmega ,(\xi _{1},\xi _{2})\in R^{N+1}, $$where \(F(x,t,\xi _{1},\xi _{2})=\int _{0}^{t} f(x,s,\xi _{1},\xi _{2})\,ds\).
It is well known that (AR) is a important technical condition to apply the mountain pass theorem. This condition implies that
If \(f(x,u)\) is asymptotically linear at \(u=0\) or \(u=+\infty \). then \(f(x,u)\) does not satisfy the (AR) condition. In [14], A. Bensedik and M. Bouchekif considered second-order elliptic equations of Kirchhoff type with an asymptotically linear potential
On the other hand, the classical equation involving a biharmonic operator
has been extensively studied using the mountain pass theorem when \(a(x)\equiv 0\) and \(f(x,u)\) is asymptotically linear at \(u=0\) or \(u=+\infty \). We refer the reader to [15, 16]. In particular, in [17], Pu et al. considered problem (1.2) when \(a(x)\neq 0\).
Until now, there are few works on problem (1.1) when \(a(x)\neq 0\) and \(f(x,u)\) does not satisfy the (AR) condition. Inspired by these references, in this paper, we discuss the existence and multiplicity of solutions of problem (1.1) when \(a(x)\neq 0\) and the nonlinearity f is asymptotically linear at \(u=0\) or \(u=+\infty \).
2 Preliminaries
Assume that the function \(m(t)\) satisfies the following conditions:
- \((M)\):
-
\(m: R^{+}\rightarrow R^{+}\) is continuous, nondecreasing, and there exists \(m_{1} \geq m_{0}>0\) such that
$$ m_{0}=\min_{t\in R^{+}}m(t)=m(0),\quad \quad m_{1}=\sup_{t\in R^{+}}m(t). $$
Remark
In [14] and [18], the function \(m(t)\) is assumed that satisfy \((M)\) and there exits \(t_{0} > 0\) such that \(m (t) = m_{1}\), \(\forall t > t_{0} \).
First, we study the nonlinear eigenvalue problem
Let \((\lambda _{k},\phi _{k})\) be the eigenvalue and the corresponding eigenfunction of \((-\Delta ,H_{0}^{1}(\varOmega ))\), namely
Set
Via some simple computations, we get
Set
and so \(\varLambda _{k}\) (\(k=1,2,\ldots \)) are the eigenvalues of the operator L associated to the eigenfunction \(\phi _{k}\).
Assume that the eigenfunctions \(\phi _{k}\) are suitably normalized with respect to the \(L^{2}(\varOmega )\) inner product, namely
Expression (2.1) can be rewritten as
For each eigenvalue \(\lambda _{k}\) being repeated as often as multiplicity, recall that
and if \((M)\) holds, then
Denote
then we know that
It is well known that
Similarly, we have
Lemma 2.1
Assume that\((M)\)holds, then
Proof
Denote
then it is clear that
Let \(u_{0}\in {\mathbf{H}}^{2}(\varOmega )\cap {\mathbf{H}}^{1}_{0}(\varOmega )\) achieve \(\varLambda _{0}\), then \(\int _{\varOmega } \vert u_{0} \vert ^{2} \,dx=1\), \(\int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx \geq \lambda _{1} \) and \(u_{0}=0\) on ∂Ω, therefore
which implies that
then
So \(\varLambda _{0} = \varLambda _{1}\).
Let \({\mathbf{H}}={\mathbf{H}}^{2}(\varOmega )\cap {\mathbf{H}}^{1}_{0}( \varOmega )\) be the Hilbert space equipped with the standard inner product
and the deduced norm
It is well know that \(\Vert u \Vert _{H} \) is equivalent to \((\int _{\varOmega } \vert \Delta u \vert ^{2} \,dx)^{\frac{1}{2}}\). And there exists \(\tau >0\) such that
Denote
and
It is obvious that the norms \(\Vert u \Vert \) and \(\Vert u \Vert _{m_{0}}\) are equivalent to the norm \(\Vert u \Vert _{H} \) in H. And since \(m_{0}< m_{1}\), we have
where \(\theta =\frac{m_{0}}{m_{1}}\in (0.1)\).
Throughout this paper, we denote by C universal positive constants, unless otherwise specified, and
By the Sobolev embedding theorem, there is a positive \(K_{q}\) such that
Specially, when condition \((M)\) holds and \(q=2\), by Lemma 2.1, then
The mountain pass theorem and the Ekeland variational principle are our main tools, which can be found in [19]. □
Lemma 2.2
LetEbe a real Banach space, and\(I\in C^{1}(E, R)\)satisfy (PS) condition. Suppose
-
1
There exist\(\rho >0\), \(\alpha >0\)such that
$$ I| _{\partial B_{\rho }}\geq I(0)+\alpha , $$where\(B_{\rho }=\{ u\in E| \Vert u \Vert \leq \rho \}\).
-
2
There is an\(e\in E\)with\(\Vert e \Vert >\rho \)such that
$$ I(e)\leq I(0). $$
Then\(I(u)\)has a critical valuecwhich can be characterized as
where\(\varGamma =\{\gamma \in C([0,1],E)| \gamma (0)=0,\gamma (1)=e\}\).
Lemma 2.3
LetVbe a complete metric space and\(I: V\rightarrow R\cup \{+\infty \}\)be lower semicontinuous, bounded from below. Let\(\varepsilon >0\)be given and\(v\in V\)be such that
Then there exists\(u\in V\)such that
and for all\(w\neq u\)in V,
3 Main results
A function \(u\in \mathbf{H}\) is called a weak solution of (1.1) if
holds for any \(v\in \mathbf{H}\). Let \(J:\mathbf{H}\rightarrow R\) be the functional defined by
where
It is easy to see that \(J\in C^{1}({\mathbf{H}}, R)\) and the critical points of J in H correspond to the weak solutions of problem (1.1).
We make the following assumptions.
- \((A)\):
-
\(a(x)\in {\mathbf{C}}(\overline{\varOmega })\), \(a(x)\geq 0\), \(\forall x\in \overline{\varOmega }\) and \(\Vert a(x) \Vert _{\infty }=\bar{a}>0\);
- \((F_{0})\):
-
\(tf(x,t)\geq 0\) for \(x\in \overline{{\varOmega }}\), \(t\in {\mathbf{R}}\);
- \((F_{1})\):
-
\(\lim_{ \vert t \vert \rightarrow 0}\frac{f(x,t)}{t}=p(x)\) uniformly a.e. \(x\in {\varOmega }\), where \(0< p(x)\in L^{\infty }(\varOmega )\), and \(\Vert p(x) \Vert _{\infty }<\theta \varLambda _{1}\);
- \((F_{2})\):
-
\(\lim_{ \vert t \vert \rightarrow +\infty }\frac{f(x,t)}{t}=l\) (\(-\infty < l< + \infty \)) uniformly a.e. \(x\in {\varOmega }\).
Our first main result is concluded as the following theorem:
Theorem 3.1
Assume the function\(m(t)\)satisfies\((M)\), \(a(x)\)satisfies\((A)\), and the nonlinearity\(f(x,t)\)satisfies\((F_{1})\)and\((F_{2})\), then problem (1.1) has at least one solution if\(l< \varLambda _{1}\).
Proof
It is easy to see, from condition \((F_{1})\), that \(f(x,0)= 0\) for \(x\in \varOmega \). So \(u=0\) is the trivial solution of (1.1). From condition \((F_{2})\), we can take \(\varepsilon =\frac{1}{2}(\varLambda _{1}-l)>0\), and there exists \(T>0\) such that
for all \(\vert t \vert \geq T\) and a.e. \(x\in \varOmega \). By the continuity of F, there exists \(C>0\) such that
for all \((x,t)\in \varOmega \times R\). On the other hand, from \((M)\) it follows that
Then we have
which shows that J is coercive. Moreover, conditions \((F_{1})\) and \((F_{2})\) imply that J is weakly lower semicontinuous in H. Therefore we get a global minimum \(u_{1}\) of J.
Next, we prove \(u_{1}\neq 0\), so it is a nontrivial solution of (1.1). From condition \((F_{1})\), there exists \(C>0\) such that
for all \(\vert t \vert \) small enough and \(x\in \varOmega \). It follows that
for all \(\vert t \vert \) small enough and \(x\in \varOmega \). From condition \((A)\), we can chose \(v\in {\mathbf{H}}\) such that
Then we have
Therefore, we get that \(J(u_{1})<0\). It is clear that \(J(0)=0\). Thus, \(u_{1}\) is a nontrivial solution of (1.1). □
Our second result is the following theorem:
Theorem 3.2
Assume the function\(m(t)\)satisfies\((M)\), \(a(x)\)satisfies\((A)\), and the nonlinearity\(f(x,t)\)satisfies\((F_{0})\), \((F_{1})\), and\((F_{2})\), then there exists a positive constant\(a_{0}\)such that problem (1.1) has at least three nontrivial solutions if\(\bar{a}< a_{0}\)and\(\bar{\varLambda }_{1}< l<+\infty \).
Before proving Theorem 3.2, we give two lemmas.
Lemma 3.1
Suppose the conditions of Theorem 3.2hold, then there exists a positive constant\(a_{0}\)such thatJsatisfies the following conditions for\(\bar{a}< a_{0}\)and\(\bar{\varLambda }_{1}< l<+\infty \):
-
1.
There exist constants\(\rho >0\), \(\alpha >0\)such that\(J| _{\partial B_{\rho }}\geq \alpha \)with\(B_{\rho }=\{ u\in {\mathbf{H}} : \Vert u \Vert \leq \rho \}\);
-
2.
\(J(t\varphi _{1})\rightarrow -\infty \)as\(t\rightarrow +\infty \).
Proof
(Claim 1) By \((F_{1})\) and \((F_{2})\), there exists \(C>0\) such that for all \((x,t)\in \varOmega \times R\) and \(p\in (1,\frac{N+4}{N-4})\), we have
From inequalities (2.2), (2.3) and (3.1), we have
Setting
when \(\bar{a}\leq a_{0}\) and \(\Vert u \Vert =\rho \), it follows that
So, Claim 1 is proved.
(Claim 2) By \((F_{2})\) and for \(l>\bar{\varLambda }_{1}\), there exists \(C>0\) such that
for all \((x,t)\in \varOmega \times R\). Let \(\lambda _{1}\) and \(\phi _{1}\) be the first eigenvalue and eigenfunction of \((-\Delta ,H_{0}^{1}(\varOmega ))\) with \(\int _{\varOmega } \vert \phi _{1} \vert ^{2} \,dx=1\). We know that
Then, we have
Hence, \(J(t\psi _{1}) \rightarrow -\infty \), \(t\rightarrow +\infty \).
The proof of Lemma 3.1 is completed. □
Let
and
Define functionals \(J^{\pm }: \mathbf{H} \rightarrow \mathbf{R}\) as follows:
where \(F^{\pm }(t)=\int _{0}^{t}f^{\pm }(x,s)\,ds\).
Lemma 3.2
Assume that\((M)\), \((A)\)and\((F_{0})\)–\((F_{2})\)hold, and\(\bar{\varLambda }_{1} < l < +\infty \), then\(J^{\pm }(u)\)satisfies the (PS) condition.
Proof
We just prove that \(J^{+}(u)\) satisfies the (PS) condition. The proof for \(J^{-}(u)\) is similar. Let \(\{u_{n}\}\in \mathbf{H}\) be a (PS) sequence, namely
Firstly, we claim that \(\{u_{n}\}\) is bounded in H. If not, we may assume that \(\Vert u_{n} \Vert \rightarrow +\infty \) as \(n\rightarrow +\infty \). Let \(w_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert }\), then \(\Vert w_{n} \Vert =1\). Passing to a subsequence, we may assume that there exists \(w\in {\mathbf{H}}\) such that
By \((F_{1})\) and \((F_{2})\), we see that there exist \(C_{1}\) and \(C_{2}\) such that
for all \((x,t)\in \varOmega \times {\mathbf{R}}\) and define
Then we claim that \(w\neq 0\). Otherwise, if \(w\equiv 0\), we know that \(w_{n}\rightarrow 0\) strongly in \({\mathbf{L}}^{r}(\varOmega )\). Dividing (3.2) by \(\Vert u_{n} \Vert ^{2}\), we have
It follows from (3.1) and (3.5) that
which is impossible, so \(w\neq 0\).
Let us define
Then, for all \(v\in {\mathbf{H}}\), we have
So,
where \(w^{+}(x)=\max {\{w(x), 0\}}\). On the other hand, since \(\Vert u_{n} \Vert \rightarrow +\infty \), we have \(\vert u_{n}(x) \vert = \Vert u_{n} \Vert \vert w_{n}(x) \vert \rightarrow +\infty \) for \(x\in \varOmega _{1}\). Therefore, by \((F_{2})\) and the dominated convergence theorem, we get
Combining (3.6) and (3.7), we obtain
Now, (3.3) implies that, for all \(v\in {\mathbf{H}}\), we have
Dividing by \(\Vert u_{n} \Vert \), we get
Since
as \(n\rightarrow +\infty \), we can suppose that there exists a subsequence, still denoted \(\{\int _{\varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\}\), such that
otherwise, there exists \(K>0\) such that
and furthermore, there exist a subsequence, still denoted \(\{\int _{\varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\}\), and a constant \(t'\geq 0\) such that
In case (3.10) holds, by \((M)\), we have
Combining (3.4), (3.8), (3.9) and (3.10), as \(n\rightarrow +\infty \), we obtain
Taking \(v=\phi _{1}\) in (3.13), we have
Noticing that \(\phi _{1}\) is the positive solution of
we have
Thus, from (3.14) and (3.15), we get
If \(w(x)\geq 0\) a.e. in Ω, since \(w(x)\neq 0\), we have \(\int _{\varOmega }w\phi _{1} \,dx>0\). Then (3.15) implies that
which contradicts \(l>\bar{\varLambda }_{1}\). Otherwise, let \(\varOmega _{-}=\{x\in \varOmega| w(x)<0\}\) and suppose \(\vert \varOmega _{-} \vert >0\). Then \(\int _{\varOmega _{-}}-w\phi _{1} \,dx>0\) and \(\int _{\varOmega }w^{+}\phi _{1} \,dx>\int _{\varOmega }w\phi _{1} \,dx>0\). It follows from (3.15) again that
which contradicts \(l>\bar{\varLambda }_{1}\).
So \(\{u_{n}\}\) is bounded in X.
In case (3.11) holds, by \((M)\), we have
Combining (3.4), (3.8), (3.9) and (3.17), as \(n\rightarrow +\infty \), we obtain
Taking \(v=\phi _{1}\) in (3.18), we have
Notice that \(\phi _{1}\) is also the positive solution of
where \(\varLambda '_{1}=\lambda _{1}^{2}+m'\lambda _{1}\). Then we have
From (3.19) and (3.20), we get
Notice that for \(\varLambda '_{1}\leq \bar{\varLambda }_{1}\), similar to the discussions in case (3.10) holds, (3.21) implies a contradiction to \(l>\bar{\varLambda }_{1}\).
So \(\{u_{n}\}\) is bounded in X.
Now, since Ω is bounded and \((F_{1})\), \((F_{2})\) hold, by using the Sobolev embedding theorem and the standard procedures, we can easily prove that \(\{u_{n}\}\) has a convergent subsequence. The proof of the lemma is completed. □
Proof of Theorem 3.2.
From the proof of Lemma 3.1, it is easy to see that \(J^{+}(u)\) and \(J^{-}(u)\) satisfy the conditions of Lemma 3.1. So there exist \(\rho >0\), \(\alpha >0\), and \(e\in {\mathbf{H}}\) with \(\Vert e \Vert >\rho \) such that
It is clear that \(J^{\pm }(0)=0\). Moreover, by Lemma 3.2, the functionals \(J^{\pm }\) satisfy the (PS) condition. By Lemma 2.2, we know that \(J^{\pm }\) has the critical value \(c^{\pm }\), respectively, which can be characterized as
where \(\varGamma =\{\gamma \in C([0,1],{\mathbf{H}})| \gamma (0)=0,\gamma (1)=e \}\). So there exist critical points \(u_{1}, u_{2} \in {\mathbf{H}}\) such that
Since \(f^{+}(x,t)\geq 0\) and \(f^{-}(x,t)\leq 0\), by the comparison principles for some fourth order elliptic problems [20], \(u_{1}\) is a positive solution of (1.1) and \(u_{2}\) is a negative solution of (1.1).
Next, we prove that problem (1.1) has another solution \(u_{3} \in {\mathbf{H}}\) such that \(J(u_{3})<0\). For \(\rho >0\) given by Lemma 3.1, define \(B_{\rho }=\{u\in E: \Vert u \Vert \leq \rho \}\) and then \(B_{\rho }\) is a complete metric space with the distance \(\operatorname{dist}(u,v)= \Vert u-v \Vert \) for \(u,v\in B_{\rho }\). By Lemma 3.1, we know that
Clearly, \(J\in C^{1}(B_{\rho },R)\), so J is bounded from below on \(B_{\rho }\). And we know that J is lower semicontinuous.
Similar to the proof of Theorem 3.1, there exists \(v \in {\mathbf{H}}\) such that
Then letting \(c_{1}=\inf \{J(u):u\in B_{\rho }\}\), we get that \(c_{1}<0\). By Lemma 2.3, for any \(k>0\), there is a \(\{u_{k}\}\) such that
Now we claim that \(\Vert u_{k} \Vert <\rho \) for k large enough. Otherwise, if \(\Vert u_{k} \Vert =\rho \) for infinitely many k, and, without loss of generality, we may suppose that \(\Vert u_{k} \Vert =\rho \) for all \(k>1\). It follows from (3.22) that \(J(u_{k})\geq \alpha >0\). Letting \(k\rightarrow \infty \), we see that \(0>c_{1}\geq \alpha >0\), which is a contradiction.
For any \(u\in E\) with \(\Vert u \Vert =1\), let
for any fixed \(k\geq 1\). We get
so \(w_{k}\in B_{\rho }\) for \(t>0\) small enough. It follows from Lemma 2.3 that
Thus, we have
and
Then \(\vert J'(u_{k}) \vert \leq \frac{1}{k}\rightarrow 0\) and \(J(u_{k})\rightarrow c_{1}\) as \(k\rightarrow \infty \). Therefore \(\{u_{k}\}\) is a (PS) sequence at level \(c_{1}\). From Lemma 3.2, \(\{u_{k}\}\) has a convergent subsequence. Hence, we see that there exists \(u_{3}\in {\mathbf{H}}\) such that \(J'(u_{3})=0\) and \(J(u_{3})=c_{1}<0\). Thus, \(u_{3}\) is a nontrivial weak solution of (1.1) and \(u_{3}\neq u_{1}\), \(u_{3}\neq u_{2}\). □
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This work was supported by the Fundamental Research Funds for the Central Universities (2632020PY02) and the National Natural Foundation of China-NSAF (Grant No. 11571092).
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Yuanfang, R., Yukun, A. Existence and multiplicity of solutions for nonlocal fourth-order elliptic equations with combined nonlinearities. Bound Value Probl 2020, 130 (2020). https://doi.org/10.1186/s13661-020-01430-4
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DOI: https://doi.org/10.1186/s13661-020-01430-4