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

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.