# 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.