A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition

Khadijah Sharaf

doi:10.3934/dcds.2017070

Article Contents

2017, Volume 37, Issue 3: 1691-1706. Doi: 10.3934/dcds.2017070

This issue Previous Article Dynamical canonical systems and their explicit solutions Next Article Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential

A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition

Khadijah Sharaf^1,

Department of Mathematics, King Abdulaziz University, P.O. 80230, Jeddah, Kingdom of Saudi Arabia

Received: June 2016

Revised: October 2016

Published: May 2017

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Abstract

In this paper we consider the following nonlinear critical problem: $-Δ u= (1+\varepsilon_0 K_0(x)) u^\frac{n+2}{n-2}$ , $u>0$ in $Ω$ , $u=0$ , on $\partial Ω$ , where $Ω$ is a bounded domain of $\mathbb{R}^n$ , $K_0$ is a given function and $\varepsilon_0$ is a small parameter. Under the assumption that $K_0$ is flat near its critical points, we prove an existence result in terms of the Euler-Hopf index. We believe that it is the very first result in this direction that we do not need any restrictions on the flatness coefficient.

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Mathematics Subject Classification: 35J65, 35J20, 35J60.

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