# 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

### Downloads