We consider the supercritical problem -Delta u = vertical bar u vertical bar(p-2) u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, N >= 3, and p >= 2* := 2N/N-2. Bahri and Coron showed that if Omega has nontrivial homology this problem has a positive solution for p = 2*. However, this is not enough to guarantee existence in the supercritical case. For p >= 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 <= k <= N - 3 there are bounded smooth domains in 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
We consider the supercritical problemwhere Θ is a bounded smooth domain in R N , N ≥ 3, p > 2 * := 2N N−2 , and Θǫ is obtained by deleting the ǫ-neighborhood of some sphere which is embedded in Θ. In some particular situations we show that, for ǫ > 0 small enough, this problem has a positive solution vǫ and that these solutions concentrate and blow up along the sphere as ǫ → 0.Our approach is to reduce this problem to a critical problem of the formin a punctured domain Ωǫ := {x ∈ Ω : |x − ξ 0 | > ǫ} of lower dimension, by means of some Hopf map. We show that, if Ω is a bounded smooth domain in R n , n ≥ 3, ξ 0 ∈ Ω, Q ∈ C 2 (Ω) is positive and ∇Q(ξ 0 ) = 0 then, for ǫ > 0 small enough, this problem has a positive solution uǫ, and that these solutions concentrate and blow up at ξ 0 as ǫ → 0.Key words: Nonlinear elliptic problem; supercritical problem; nonautonomous critical problem; positive solutions; domains with a spherical perforation, blow-up along a sphere.MSC2010: 35J60, 35J20.
We show that in every dimension N ≥ 3 there are many bounded domains Ω ⊂ R N , having only finite symmetries, in which the Bahri-Coron problem −∆u = |u| 4/(N−2) u in Ω, u = 0 on ∂Ω, has a prescribed number of solutions, one of them being positive and the rest sign changing. Date: February 2012.
We study the weakly coupled critical elliptic systemwhere Ω is a bounded smooth domain in R N , N ≥ 3, 2 * := 2N N−2 is the critical Sobolev exponent, µ1, µ2 > 0, α, β > 1, α + β = 2 * and λ ∈ R.We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on Ω, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as λ → −∞.We also obtain existence of infinitely many solutions to this system in Ω = R N .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.