SUPERCRITICAL PROBLEMS IN DOMAINS WITH THIN TOROIDAL HOLES
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2014
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Abstract
In this paper we study the Lane-Emden-Fowler equation
(P)(epsilon) {(Delta)u+vertical bar u vertical bar(q-2) u = 0 in D-epsilon,D- u = 0 on partial derivative D-epsilon.
Here D-c = D\ {x epsilon D : dist (x, Gamma(l) ) <= epsilon }, D is a smooth bounded domain in R-N, Gamma(l) is an l-dimensional closed manifold such that Gamma l subset of D with 1 <= l <= N - 3 and q = 2(N - l)/ N-l-2. We prove that, under some symmetry assumptions, the number of sign changing solutions to (P)(epsilon), increases as goes to zero.
(P)(epsilon) {(Delta)u+vertical bar u vertical bar(q-2) u = 0 in D-epsilon,D- u = 0 on partial derivative D-epsilon.
Here D-c = D\ {x epsilon D : dist (x, Gamma(l) ) <= epsilon }, D is a smooth bounded domain in R-N, Gamma(l) is an l-dimensional closed manifold such that Gamma l subset of D with 1 <= l <= N - 3 and q = 2(N - l)/ N-l-2. We prove that, under some symmetry assumptions, the number of sign changing solutions to (P)(epsilon), increases as goes to zero.
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Supercritical problem, concentration on l-dimensional manifolds