Sign changing solutions to a Bahri-Coron's problem in pierced domains
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Date
2008
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AMER INST MATHEMATICAL SCIENCES-AIMS
Abstract
We consider the problem
{-Delta u = vertical bar u vertical bar(4/N-2u) in Omega/{B(xi(1),epsilon) boolean OR B(xi(2),epsilon)}, u = 0 on partial derivative(Omega\{B(xi(1),epsilon) boolean OR B(xi(2),epsilon)},
where Omega is a smooth bounded domain in R-N, N >= 3, xi(1), xi(2) are different points in Omega and epsilon is a small positive parameter. We show that, for epsilon small enough, the equation has at least one pair of sign changing solutions, whose positive and negative parts concentrate at xi(1) and xi(2) as epsilon goes to zero.
{-Delta u = vertical bar u vertical bar(4/N-2u) in Omega/{B(xi(1),epsilon) boolean OR B(xi(2),epsilon)}, u = 0 on partial derivative(Omega\{B(xi(1),epsilon) boolean OR B(xi(2),epsilon)},
where Omega is a smooth bounded domain in R-N, N >= 3, xi(1), xi(2) are different points in Omega and epsilon is a small positive parameter. We show that, for epsilon small enough, the equation has at least one pair of sign changing solutions, whose positive and negative parts concentrate at xi(1) and xi(2) as epsilon goes to zero.
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Keywords
critical Sobolev exponent, pierced domain, single blow up, sign changing solutions, CRITICAL SOBOLEV EXPONENTS, SEMILINEAR ELLIPTIC-EQUATIONS, MINIMAL NODAL SOLUTIONS, CRITICAL GROWTH, VARIATIONAL PROBLEM, SYMMETRIC DOMAIN, EXISTENCE, TOPOLOGY, HOLES, COMPACTNESS