    October  2012, 32(10): 3567-3585. doi: 10.3934/dcds.2012.32.3567

## Perturbed elliptic equations with oscillatory nonlinearities

 1 School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China 2 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received  August 2010 Revised  February 2012 Published  May 2012

In this paper, arbitrarily many solutions, in particular arbitrarily many nodal solutions, are proved to exist for perturbed elliptic equations of the form \begin{equation*}\label{} \left\{ \begin{array}{ll} \displaystyle -\Delta_p u+|u|^{p-2}u = Q(x)(f(u)+\varepsilon g(u)),\ \ \ x\in \mathbb R^N, \\ u\in W^{1,p}(\mathbb R^N), \end{array} \right. (P_\varepsilon) \end{equation*} where $\Delta_p$ is the $p$-Laplacian operator defined by $\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $Q\in \mathcal{C}(\mathbb R^N,\mathbb R)$ is a positive function, $f\in\mathcal{C}(\mathbb R, \mathbb R)$ oscillates either near the origin or near the infinity, and $\epsilon$ is a real number. For $g$ it is only required that $g\in\mathcal{C}(\mathbb R, \mathbb R)$. Under appropriate assumptions on $Q$ and $f$ the following results which are special cases of more general ones are proved: the unperturbed problem $(P_0)$ has infinitely many nodal solutions, and for any $n\in\mathbb N$ the perturbed problem $(P_\varepsilon)$ has at least $n$ nodal solutions provided that $|\epsilon|$ is sufficiently small.
Citation: Zuji Guo, Zhaoli Liu. Perturbed elliptic equations with oscillatory nonlinearities. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3567-3585. doi: 10.3934/dcds.2012.32.3567
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##### References:
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