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

Abstract

We study the problem% \[ -Δv+λv=| v| ^{p-2}v\text{ in }Ω,\text{\qquad}v=0\text{ on $\partialΩ$},\text{ }% \] for $λ\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form% \[ Ω:=\{(y,z)\in\mathbb{R}^{N-m-1}\times\mathbb{R}^{m+1}:(y,| z| )\inΘ\}, \] where $m\geq1,$ $N-m\geq3,$ and $Θ$ is a bounded domain in $\mathbb{R}% ^{N-m}$ whose closure is contained in $\mathbb{R}^{N-m-1}\times(0,\infty)$. Under some symmetry assumptions on $Θ$, we show that this problem has infinitely many solutions for every $λ$ in an interval which contains $[0,\infty)$ and $p>2$ up to some number which is larger than the $(m+1)^{st}$ critical exponent $2_{N,m}^{\ast}:=\frac{2(N-m)}{N-m-2}$. 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 $p\rightarrow2_{N,m}^{\ast}$ 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% \[ -Δu=| u| ^{2_{n}^{\ast}-2}u,\text{\qquad}u\in D^{1,2}(\mathbb{R}^{n}), \] where $2_{n}^{\ast}:=\frac{2n}{n-2}$ is the critical exponent in dimension $n.$\medskip