Asymptotics for a high-energy solution of a supercritical problem

TL;DR

This paper investigates the asymptotic behavior of high-energy solutions to a supercritical p-Laplacian problem with Neumann boundary conditions, revealing that solutions converge to a constant as the nonlinearity exponent grows large.

Contribution

It provides a detailed analysis of the limit profile of higher energy solutions for a supercritical p-Laplacian problem, including new a priori estimates and convergence results.

Findings

Higher energy solutions converge to the constant 1 as q→∞

The minimal energy solution behaves differently from higher energy solutions

New estimates and tools for analyzing supercritical problems are developed

Abstract

In this paper we deal with the equation \[-\Delta_p u+|u|^{p-2}u=|u|^{q-2}u\] for $1<p<2$ and $q>p$, under Neumann boundary conditions in the unit ball of $\mathbb R^N$. We focus on the three positive, radial, and radially non-decreasing solutions, whose existence for $q$ large is proved in [13]. We detect the limit profile as $q\to\infty$ of the higher energy solution and show that, unlike the minimal energy one, it converges to the constant $1$. The proof requires several tools borrowed from the theory of minimization problems and accurate a priori estimates of the solutions, which are of independent interest.