Abstract
In this paper, we consider the following sublinear Kirchhoff problems , in , where and with . A new sublinear growth condition is given. When is not odd in and not integrable in , we obtain the existence of solutions for the above problem.
1. Introduction and Main Results
In this paper, we consider the following nonlinear Kirchhoff type problems:where . The Kirchhoff type problems with general potentials on a bounded domain are introduced bywhich is related to the stationary analogue of the Kirchhoff equation With the development of the variational methods in the last decades, many mathematicians tried to use these methods to search for the existence and multiplicity of solutions for differential equations (see [1–30]). After the work of Lions [14], the Kirchhoff problems have been studied by many mathematicians using the functional analysis methods.
When is superlinear but subcritical with respect to , many mathematicians obtained the existence and multiplicity of solutions for problem (1) (see [4, 9, 15–18, 20, 22–24, 30]). But there are only a few papers concerning the sublinear Kirchhoff type problems. In 2013, Ye and Tang [30] obtained the existence of infinitely many solutions for (1) by symmetrical mountain pass theorem with , where and . In [2], Bahrouni obtained the infinitely many solutions for sublinear Kirchhoff equations with sign-changing potentials. In 2014, Duan and Huang [6] obtained the existence and multiplicity of solutions for problem (1) with more general sublinear nonlinearities. In a recent paper [19], Li and Zhong showed the existence of infinitely many solutions for problem (1) with local sublinear nonlinearities. However, in the above papers, the nonlinear term is assumed to belong to some with respect to for some . Under a coercive condition, Wang and Han considered a class of sublinear nonlinearities for problem (1) and showed the existence of infinitely many solutions when is odd in which is the following theorem.
Theorem 1 (see [25]). Suppose that the following conditions are satisfied:
and there are constants and such that, for any , , where denotes the Lebesgue measure;
there exists a constant such that and for all and ;
There is a ball such that and where ;
There is a constant and a function such that for all and .
Then Equation (1) possesses a sequence of weak solutions in such that in as .
With coercive condition , the authors obtained a new compact embedding theorem, stated as follows.
Lemma 2 (see [25]). Suppose that satisfies conditions . Then is compact embedded into for any .
Remark 3. From Lemma 2, for any , there exists a constant such that
A natural question is whether there exists solution for problem (1) if there is no symmetrical condition on and is not integrable in . In this paper, we try to give an existence theorem on this problem. Motivated by the above papers, we consider problem (1) with some new sublinear nonlinearities and, before we state our result, we assume that is a continuous function space such that, for any , there exists a constant such that
(i) for all ; (ii) as .
In order to obtain the critical points of the corresponding functional, we consider the following function space in this paper: with the inner product For any , it follows from (6) that In order to show that the infimum can be achieved, consider a minimizing sequence such that It is easy to see that there exists such that . From Brézis-Lieb Lemma and Lemma 2, we obtain , andThen we state our main result.
Theorem 4. Suppose that and satisfy and the following conditions:
letting , there exist with and such that there exists , , and such that uniformly in ;
there is such that Then there exists at least one nontrivial solution for problem (1).
Remark 5. Letwhere such that is a class function. It is easy to see that satisfies . However, since , we see that does not satisfy
Remark 6. Condition is introduced by Wang and Xiao in [26] to prove the existence of periodic solutions for a class of subquadratic Hamiltonian systems. As we know, this is the first time to use such condition on the existence of solutions for sublinear Kirchhoff equations.
2. Preliminaries
Lemma 7. Suppose that holds; then there exists such that
Proof. For any and , let . It follows from condition thatfor all . Set Then yields that For any , (17) shows that , which implies that Then we can obtainFrom (21) and , we can obtain our conclusion.
Remark 8. By the definition of , it follows from the properties of that as .
Lemma 9. Suppose that satisfies conditions , , and ; then, for any , there exists such thatand
Proof. From Lemma 7, there exists such that For , it follows from thatWe can easily obtain (22) from and (25). (23) can be deduced from (22) directly.
3. Proof of Theorem 1
By standard arguments, we know that the functional , defined byis well defined and the critical points of are the solutions for problem (1).
Lemma 10. Suppose that , hold; then satisfies the condition.
Proof. From Remark 8 and , there exists such thatFirst, we show that is bounded in . It follows from the definition of , (16) and (27), thatwhich implies that is bounded from below on . Since we have Lemma 2, it follows from a standard argument, we obtain that as . Hence satisfies the condition.
Proof of Theorem 1. By above discussions, we can see that is of class and satisfies the condition. Similar to (28), we obtain that is bounded from below. Then is a critical value of and there exists such that . Finally, we show that . With being small enough, it follows from and (11) thatThen we can deduce , which implies that .
Abbreviations
| condition: | Palais-Smale condition. |
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The research and writing of this manuscript were a collaborative effort from both of the authors. Zhuo Yao and Wei Yang discussed many details of the problems together. Yao managed this manuscript and Yang revised it. Both of the authors read and approved the final version of the manuscript.
Acknowledgments
The financial support of National Science and Technology Support Plan Projects of China (no. 2014BAB02B03) is gratefully acknowledged.