Positive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents

Abstract

We study the problem −Δv + λv = |u|p−2 u in Ω, u= 0 on ∂Ω, for λ ∈ R and supercritical exponents p, in domains of the form Ω := {(y, z) ∈ RN−m−1 x Rm+1 : (y, |z|) ∈ Θ}, where m ≥ 1, N − m ≥ 3, and Θ is a bounded domain in RN−m whose closure is contained in RN−m−1 x (0,∞). Under some symmetry assumptions on Θ, we show that this problem has infinitely many solutions for every λ in an interval which contains [0,∞) and p > 2 up to some number which is larger than the (m+1)st critical exponent 2∗N,m := 2(N−m)/N−m−2 . We also exhibit domains with a shrinking hole, in which there are a positive and a nodal solution which concentrate on a sphere, developing a single layer that blows up at an m dimensional sphere contained in the boundary of Ω, as the hole shrinks and p → 2∗N,m from above. The limit profile of the positive solution, in the transversal direction to the sphere of concentration, is a rescaling of the standard bubble, whereas that of the nodal solution is a rescaling of a nonradial sign-changing solution to the problem −Δu = |u|2∗n−2 u, u ∈ D1,2(Rn), where 2∗n := 2n n−2 is the critical exponent in dimension n.