Infinitely many large energy solutions for Schrödinger-Kirchhoff type problem in \(R^N\)

Volume 9, Issue 2, pp 652--660 http://dx.doi.org/10.22436/jnsa.009.02.28

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Authors

Bitao Cheng - School of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, P. R. China. - School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P. R. China. Xianhua Tang - School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, P. R. China.

Abstract

In this paper, we consider the following Schrödinger-Kirchhoff-type problem \[ \begin{cases} -(a+b\int_{R^N}|\nabla u|^2 dx)\Delta u+V(x)u=g(x,u) \,\,&\hbox{for} \,\,x\in R^N,\qquad (1.1)\\ u(x)\rightarrow 0 \,\,&\hbox{as} \,\,|x|\rightarrow\infty, \end{cases} \] where constants \(a > 0; b \geq 0, N = 1; 2\) or \(3, V \in C(R^N;R), g \in C(R^N \times R;R)\). Under more relaxed assumptions on \(g(x; u)\), by using some special techniques, a new existence result of infinitely many energy solutions is obtained via Symmetric Mountain Pass Theorem.

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Keywords

• Schrödinger-Kirchhoff type problem
• critical point
• symmetric Mountain Pass Theorem
• variational methods.

MSC

•  35J20
•  35J60

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