Abstract
In this paper, we study the following nonlocal problem where are constants, , , is a positive function, and is a smooth bounded domain in with . By variational methods, we obtain a pair of nontrivial solutions for the considered problem provided is small enough.
1. Introduction and Main Results
This paper is concerned with the existence and multiplicity of nontrivial solutions for the following nonlocal problem with Dirichlet boundary value conditions: where , , , is a positive function, and is a smooth bounded domain in with .
In the past two decades, the following Kirchhoff type problems with Dirichlet boundary value conditions have attracted great attention of many researchers. Such problems are often viewed as nonlocal because of the appearance of the term , which implies that (2) is no longer a pointwise identity. It is worthwhile pointing out that the equation in (2) arises in various models of physical and biological systems. Indeed, problem (2) is related to the stationary analogue of the following equation: which was first presented by Kirchhoff [1] as an extension of the classical d’Alembert wave equation for free vibrations of elastic strings, where denotes the lateral displacement, the mass density, the initial tension, the cross-section area, the Young modulus of the material, and the length of the string. Under different assumptions on , many interesting results on the existence of solutions to (2) were obtained. We refer the interested readers to [2–14] and the references therein.
However, we now face a new nonlocal term , which is different from the well-known Kirchhoff type nonlocal term . Now, there has been some results on the existence and multiplicity of nontrivial solutions to this new nonlocal problem (see [15–22]).
In particular, Yin and Liu [15] firstly studied this kind of problem: where , and obtained the existence and multiplicity of solutions for the problem.
In [16], Lei et al. considered where , and proved under certain condition on , that there are at least two positive solutions. After this, the authors also studied the problem with singularity [17].
Wang et al. [20] investigated the nonlocal problem with critical exponent
When is a nonnegative parameter and , they showed the existence of multiple positive solutions.
Recently, Zhang and Zhang [22] studied the nonlocal problem where the parameter , is the first eigenvalue of operator , and is a superlinear function with subcritical growth. By using the Mountain Pass Theorem, the authors obtained the existence of a nontrivial solution.
As far as we know, there is no work on the existence of solution to (1), which is just our purpose here. Moreover, we extend and without assuming nonlinearity is superlinear.
Our main result can be stated as follows.
Theorem 1. Assume that , , , is a positive function, then problem (1) has at least a pair of nontrivial solutions if is small enough.
The paper is organized as follows. In Section 2, we give some notations and preliminaries. In Section 3, we prove Theorem 1 for the case of . Section 4 is devoted to the proof of Theorem 1 for the case of .
2. Notations and Preliminaries
Throughout this paper, we make use of the following notations. and are standard Sobolev spaces with the usual norm , . () denotes an open ball (the sphere) centered at with radius . and denote strong and weak convergence, respectively. For each , we denote by the best Sobolev constant for the embedding of into . Let be the sequence of eigenvalues of on satisfying , and let be the corresponding orthonormal eigenfunctions in .
By the Sobolev Theorem and , the functional is well defined on . Furthermore, it belongs to , and its critical points are precisely the weak solutions of (1). Here, we say is a weak solution to (1), if for any , it holds
Following [15], we first prove that the functional satisfies the condition for any .
Lemma 2. Under the assumptions of Theorem 1, satisfies the condition with .
Proof. Let be a sequence for with ; that is,
By the Sobolev Theorem and (10),
Since , we conclude that is bounded in . Up to a subsequence (still denoted by ), we may assume that
By using Hölder’s inequality, it follows from (12) that
as . Similarly, we also have
From the two above convergences, we get
as . We claim that is false. If, to the contrary, namely, , define a functional by
Then,
By using Hölder’s inequality again, we obtain
Hence, by using (12), we obtain
as . This shows .
On the other hand, from
and , we have .
Thus, we can deduce that
By the variational method fundamental lemma [23], we further obtain
Since , it then follows that .
By (12) and , we can use the Vitali Convergence Theorem to obtain
and consequently,
This and provide
which contradicts . Thus, the claim follows. In turn, we have from (15) that , and hence, . Combining this with the weak convergence of in , we deduce that in .☐
3. Proof of Theorem 1 for
In this section, we will use the Mountain Pass Theorem to prove the existence of a pair of nontrivial solutions of the considered problem for .
Lemma 3. Assume that . If is sufficiently small, then there is a sequence such that , and , where .
Proof. By the Sobolev Theorem, we have that
For and , the function defined by
attains its maximum value at
Take and note that, for any , , there holds
whenever
On the other hand, let , then we have that
Thus, there exists such that and .
By , we may assume that . By applying the Mountain Pass Theorem without condition [24], we construct a sequence satisfying and for
where
From easy calculations, we get
This and the definition of yield that . Thus, we complete the proof of Lemma 3.☐
Proposition 4. Assume , and is a positive function. Then, problem (1) admits a pair of nontrivial solutions if is small enough.
Proof. By Lemma 3, we obtain a sequence such that , and , provided is small enough. It then follows from Lemma 2 that there exists such that in with and , which implies that is a nontrivial nonnegative solution of (1). By the symmetry of functional , we further deduce that is a nontrivial nonpositive solution of (1). This completes the proof.☐
4. Proof of Theorem 1 for
Since , the method used in the previous section does not work here. Indeed, we shall apply the following Linking Theorem [25] to establish the existence of a pair of nontrivial solutions for problem (1) when .
Theorem 5. Let be a real Banach space with and dim. Suppose that satisfies the following:
(I1) There are such that
(I2) There are and such that with
Then, there exists a sequence satisfying and for
where
As the sequence of eigenvalues goes to infinity, there is such that . Set
Clearly, .
Lemma 6. There exists such that
Proof. Since, for any , then there exists such that where This completes the proof of Lemma 6.☐
Proposition 7. Assume , and is a positive function. Then, problem (1) admits a pair of nontrivial solutions if is small enough.
Proof. Firstly, we have for any ,
and therefore, as in the proof of Lemma 3, we can prove that satisfies the condition of Theorem 5 when is small.
Secondly, since , we also have for any ,
Moreover, if is given by Lemma 6, we can apply the equivalence of norms in the finite dimensional space, to obtain for ,
Thus, the condition is satisfied for large enough.
Finally, by Theorem 5, Lemma 6, and , we conclude that there is a sequence satisfying , and , provided is small enough. Then, we can argue as in the proof of Proposition 4 to obtain a pair of nontrivial solutions of (1). Thus, the proof of Proposition 7 is complete.☐
5. Conclusions
In this paper, we consider a new nonlocal problem different from the well-known Kirchhoff type problem. Compared with the existing literature, we extend and without assuming nonlinearity is superlinear, which causes the methods employed in the previous papers cannot be directly applied here. To overcome this difficulty, we distinguish two cases: (1) and (2) . In the former case, the existence of a nontrivial nonnegative solution is established via the Mountain Pass Theorem, while the Linking Theorem for the latter case. Furthermore, the symmetry of the corresponding functional brings us another nontrivial nonpositive solution.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (11871152).