Solutions of a pure critical exponent problem involving the halflaplacian in annularshaped domains doi:10.3934/cpaa.2011.10.1645 Abstract References Full Text (434.0K) Related Articles
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 $2^{\sharp}=2n/(n1)$ is the critical traceSobolev exponent. We assume that $\Omega$ is annularshaped, 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
Received: May 2010; Revised: April 2011; Published: May 2011. 
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