Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

Abstract

We consider the supercritical problem -Δu = |u|^{p-2}u in Ω, u=0 on \partialΩ, where $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^N,$ $N\ge 3,$ and $p\ge 2^{\ast }:=2N/(N-2).$ Bahri and Coron showed that if $\Omega$ has nontrivial homology this problem has a positive solution for $p=2^{\ast }.$ However, this is not enough to guarantee existence in the supercritical case. For $p\ge 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show that for $p>2(N-k)/(N-k-2)$ with $1\le k\le N-3$ there are bounded smooth domains in ${\mathbb{R}}^N$ whose cup-length is $k+1$ in which this problem does not have a nontrivial solution. For $N=4,8,16$ we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.

Published online in Calculus of Variations and Partial Differential Equations

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• Conformal approach to cylindrical DLA
• Solutions of semilinear elliptic equations in tubes
• Boundary clustered layers near the higher critical exponents

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• Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold
• A note on solutions of Yamabe-type equations on products of spheres
• On the solutions of a singular elliptic equation concentrating on two orthogonal spheres
• Supercritical problems in domains with thin toroidal holes
• Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces
• Ground states of critical and supercritical problems of Brezis-Nirenberg type