Perturbed elliptic equations with oscillatory nonlinearities

Zuji Guo; Zhaoli Liu

doi:10.3934/dcds.2012.32.3567

Article Contents

2012, Volume 32, Issue 10: 3567-3585. Doi: 10.3934/dcds.2012.32.3567

This issue Previous Article Existence of piecewise linear Lyapunov functions in arbitrary dimensions Next Article Transport, flux and growth of homoclinic Floer homology

Perturbed elliptic equations with oscillatory nonlinearities

Zuji Guo^1, and
Zhaoli Liu^2,

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

Received: July 31, 2010

Revised: January 31, 2012

Published: September 2012

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Abstract

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.

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Mathematics Subject Classification: 35J20, 35J65.

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