Multi-layer radial solutions for a supercritical neumann problem

TL;DR

This paper investigates the existence of multiple-layer radial solutions to a supercritical Neumann boundary value problem in the unit ball as the exponent p approaches infinity.

Contribution

It introduces new methods to prove the existence of multi-layer solutions for a supercritical nonlinear PDE with Neumann boundary conditions.

Findings

Existence of multiple-layer radial solutions as p tends to infinity.

Construction of solutions with layered structures in the supercritical regime.

Analytical techniques for handling supercritical nonlinearities.

Abstract

In this paper we study the Neumann problem\begin{equation*}\begin{cases}-\Delta u+u=u^p \& \text{ in }B\_1 \\u \textgreater{} 0, \& \text{ in }B\_1 \\\partial\_\nu u=0 \& \text{ on } \partial B\_1,\end{cases}\end{equation*}and we show the existence of multiple-layer radial solutions as $p\rightarrow+\infty$.