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Mediterranean Journal of Mathematics

, Volume 13, Issue 3, pp 1099–1116 | Cite as

A New Result on Multiplicity of Nontrivial Solutions for the Nonhomogenous Schrödinger–Kirchhoff Type Problem in R N

  • Bitao Cheng
Article

Abstract

In this paper, we consider the following nonhomogenous Schrödinger–Kirchhoff type problem
$$\left\{ \begin{array}{ll} - (a+b\int_{R^{N}}|\nabla u|^{2}dx)\triangle u + V(x)u =f(x,u)+g(x), & \,\,\,{\rm for} \, x \in R^N, \\ u(x)\rightarrow0, & \,\, {\rm as}\, |x|\rightarrow\infty,\end{array}\right.$$
(0.1)
where constants a > 0, b ≥ 0, N = 1, 2 or 3, \({V\in C(R^{N},R)}\), \({f\in C(R^{N} \times R, R)}\) and \({g\in L^{2}(R^{N})}\). Under more relaxed assumptions on the nonlinear term f that are much weaker than those in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013), using some new proof techniques especially the verification of the boundedness of Palais–Smale sequence, a new result on multiplicity of nontrivial solutions for the problem (1.1) is obtained, which sharply improves the known result of Theorem 1.1 in Chen and Li (Nonlinear Anal RWA 14:1477–1486, 2013).

Keywords

Nonhomogenous Schrödinger–Kirchhoff type problem Ekeland’s variational principle mountain Pass Theorem 

Mathematics Subject Classification

35J20 35J25 35J60 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics and Information ScienceQujing Normal UniversityQujingPeople’s Republic of China

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