Communications on Pure and Applied Analysis (CPAA)

Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains

Pages: 1645 - 1662, Volume 10, Issue 6, November 2011      doi:10.3934/cpaa.2011.10.1645

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Antonio Capella - Instituto de Matemáticas, Universidad Nacional Autónoma de Mexico, Circuito Exterior, C.U., 04510 México D.F., Mexico (email)

Abstract: We consider the nonlinear and nonlocal problem

$A_{1/2}u=|u|^{2^{\sharp}-2}u$ in $\Omega, \quad u=0$ on $\partial\Omega$

where $A_{1/2}$ represents the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions, $\Omega$ is a bounded smooth domain in $R^n$, $n\ge 2$ and $2^{\sharp}=2n/(n-1)$ is the critical trace-Sobolev exponent. We assume that $\Omega$ is annular-shaped, i.e., there exist $R_2>R_1>0$ constants such that $\{ x \in R^n$ s.t. $R_1 < |x| < R_2 \}\subset\Omega$ and $0\notin\Omega$, and invariant under a group $\Gamma$ of orthogonal transformations of $R^n$ without fixed points. We establish the existence of positive and multiple sign changing solutions in the two following cases: if $R_1/R_2$ is arbitrary and the minimal $\Gamma$-orbit of $\Omega$ is large enough, or if $R_1/R_2$ is small enough and $\Gamma$ is arbitrary.

Keywords:  Nonlinear elliptic boundary problems, multiple sign changing solutions, critical exponent, fractional Laplacian
Mathematics Subject Classification:  Primary: 35J65, 35J20; Secondary: 35S05.

Received: May 2010;      Revised: April 2011;      Available Online: May 2011.