Existence and multiplicity of periodic solutions and subharmonic solutions for a class of elliptic equations

Volume 10, Issue 12, pp 6229--6245

Publication Date: 2017-12-06

http://dx.doi.org/10.22436/jnsa.010.12.09

Authors

Xiujuan Wang - School of Mathematical Sciences, Qufu Normal University, Shandong 273165, P. R. China
Aixia Qian - School of Mathematical Sciences, Qufu Normal University, Shandong 273165, P. R. China

Abstract

This paper focuses on the following elliptic equation \[ \left\{ \begin{aligned} & -u''- p(x)u=f(x,u),\quad \text{a.e.}\quad x\in[0,l],\\ &u(0)-u(l)=u'(0)-u'(l)=0, \end{aligned} \right. \] where the primitive function of \(f(x,u)\) is either superquadratic or asymptotically quadratic as \(|u|\rightarrow\infty\), or subquadratic as \(|u|\rightarrow0\). By using variational method, e.g. the local linking theorem, fountain theorem, and the generalized mountain pass theorem, we establish the existence and multiplicity results for the periodic solution and subharmonic solution.

Keywords

Elliptic equation, periodic solution, superquadratic, subquadratic, asymptotically quadratic, subharmonic solution

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