Parameter estimates and a uniqueness result for double phase problem with a singular nonlinearity

TL;DR

This paper analyzes a double phase elliptic boundary value problem with a singular nonlinearity, providing parameter estimates, a uniqueness result, and insights into solution existence and multiplicity for large parameter values.

Contribution

It introduces an exact estimate for solutions when the parameter is large and applies it to establish a new uniqueness result for problems with singular nonlinearities.

Findings

Derived an estimate describing solution behavior for large parameters

Established a uniqueness result for a class of singular nonlinear problems

Provided tools for analyzing existence and multiplicity of solutions

Abstract

We consider the boundary value problem $-\Delta_p u_\lambda -\Delta_q u_\lambda =\lambda g(x) u_\lambda^{-\beta}$ in $\Omega$ , $u_\lambda=0$ on $\partial \Omega$ with $u_\lambda>0$ in $\Omega.$ We assume $\Omega$ is a bounded open set in $\mathbb{R}^N$ with smooth boundary, $1<p<q<\infty$, $\beta\in [0,1),$ $g$ is a positive weight function and $\lambda$ is a positive parameter. We derive an estimate for $u_\lambda$ which describes its exact behavior when the parameter $\lambda$ is large. In general, by invoking appropriate comparison principles, this estimate can be used as a powerful tool in deducing the existence, non-existence and multiplicity of positive solutions of nonlinear elliptic boundary value problems. Here, as an application of this estimate, we obtain a uniqueness result for a nonlinear elliptic boundary value problem with a singular nonlinearity.