Comparison principle for elliptic equations with mixed singular nonlinearities

Abstract

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by \begin{equation*} \begin{cases} \displaystyle -Δ_p u= \frac{f}{u^γ} + g u^q & \mbox{in $\Omega$,} \\ u = 0 & \mbox{on $\partial \Omega$,} \end{cases} \end{equation*} where $\Omega$ is an open bounded subset of ${\mathbb{R}}^N$, ${\Delta }_{p}u:=\text{div}(|\nabla u{|}^{p-2}\nabla u)$ is the usual $p$-Laplacian operator, $\gamma \ge 0$ and $0\le q\le p-1$; $f$ and $g$ are nonnegative functions belonging to suitable Lebesgue spaces.

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• How to break the uniqueness of $W_{loc}^{1,p}(\Omega )$-solutions for very singular elliptic problems by non-local terms