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

## TL;DR

This paper investigates the existence and concentration behavior of positive and nodal solutions to supercritical elliptic problems in domains with specific symmetries, revealing infinitely many solutions and solutions concentrating on spheres as parameters vary.

## Contribution

It establishes the existence of infinitely many solutions for supercritical exponents and describes solutions concentrating on spheres in domains with shrinking holes.

## Key findings

- Existence of infinitely many solutions for a range of supercritical exponents.
- Construction of solutions concentrating on spheres as the domain's hole shrinks.
- Asymptotic profiles of solutions resemble rescaled standard bubbles and sign-changing solutions.

## Abstract

We study the problem% \[ -\Delta v+\lambda v=| v| ^{p-2}v\text{ in }\Omega ,\text{\qquad}v=0\text{ on $\partial\Omega$},\text{ }% \] for $\lambda\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form% \[ \Omega:=\{(y,z)\in\mathbb{R}^{N-m-1}\times\mathbb{R}^{m+1}:(y,| z| )\in\Theta\}, \] where $m\geq1,$ $N-m\geq3,$ and $\Theta$ 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 $\Theta$, we show that this problem has infinitely many solutions for every $\lambda$ 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 $\Omega,$ 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% \[ -\Delta u=| u| ^{2_{n}^{\ast}-2}u,\text{\qquad}u\in D^{1,2}(\mathbb{R}^{n}), \] where $2_{n}^{\ast}:=\frac{2n}{n-2}$ is the critical exponent in dimension $n.$\medskip