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

We study the problem −Δv + \lambda v = |v|^{p−2} v in Ω, v = 0 on \partial Ω, for \lambda\in R and supercritical exponents p, in domains of the form Ω := {(y, z)\in R^{N−m−1} × R^{m+1} : (y, |z|) \in Θ}, where m \ge 1, N − m \ge 3, and Θ is a bounded domain in RN−m whose closure is contained in \R^{N−m−1} ×(0,1). Under some symmetry assumptions on Θ, we show that this problem has infinitely many solutions for every \lambda in an interval which contains [0,1) and p > 2 up to some number which is larger than the (m+ 1)st critical exponent 2^*_ {N,m} := \frac{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 \to 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\in D^{1,2}(\R^n), where 2^*_n := \frac{2n}{n−2} is the critical exponent in dimension n.