Asymptotic behavior of positive solutions of semilinear elliptic problems with increasing powers

TL;DR

This paper investigates the existence and asymptotic behavior of positive solutions to a class of semilinear elliptic problems with large powers, revealing different limiting behaviors as the power tends to infinity.

Contribution

It establishes existence results for two solutions of the problem with large parameters and analyzes their asymptotic limits as the power increases.

Findings

Existence of two positive solutions for large parameters

Different asymptotic behaviors as the power tends to infinity

Insights into the limit solutions as m approaches infinity

Abstract

We prove existence results of two solutions of the problem \[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{ in $\Omega$}, \\ \quad u>0 &\text{ in $\Omega$}, \\ \quad u=0 & \text{ on $\partial \Omega$}, \end{cases} \] where $L(v)=-{\rm div}(M(x)\nabla v)$ is a linear operator, $p\in (2,2^{*}]$ and $\lambda$ and $ m$ sufficiently large. Then their asymptotical limit as $m\to +\infty$ is investigated showing different behaviors.