Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 5364134, 5 pages
https://doi.org/10.1155/2019/5364134

Research Article

Existence of Solutions for Sublinear Kirchhoff Problems with Sublinear Growth

Wei Yang,1,2 Zhan Wang,³ and Zhuo Yao⁴

¹School of Insurance, Southwestern University of Finance and Economics, Chengdu, 611130, China
²Collaborative Innovation Center of Financial Security, Southwestern University of Finance and Economics, Chengdu, 611130, China
³School of Economics, Southwestern University of Finance and Economics, Chengdu, 611130, China
⁴High Temperature Materials and Magnesite Resources Engineering, University of Science and Technology Liaoning, Anshan, Liaoning 114001, China

Correspondence should be addressed to Wei Yang; 2909504391@qq.com

Received 27 February 2019; Accepted 7 April 2019; Published 21 April 2019

Academic Editor: Luigi Rodino

Copyright © 2019 Wei Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 by

which 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 , and

Then 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. Let

where 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 that

for all . Set

Then yields that

For any , (17) shows that , which implies that

Then we can obtain

From (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 that

and

Proof. From Lemma 7, there exists such that

For , it follows from that

We 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 by

is 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 that

First, we show that is bounded in . It follows from the definition of , (16) and (27), that

which 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) that

Then 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.

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