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Existence and uniqueness of solutions to some singular equations with natural growth

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Abstract

We study existence and uniqueness of nonnegative solutions to a problem which is modeled by

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = u^{-\theta }|\nabla u|^p + fu^{-\gamma }& \text {in}\, \Omega , \\ u=0 & \text {on}\ \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega\) is an open bounded subset of \({\mathbb {R}}^N\) (\(N\ge 2\)), \(\Delta _p\) is the p-Laplacian operator (\(1<p<N\)), \(f\in L^1(\Omega )\) is nonnegative and \(\theta , \gamma \ge 0\). Examples and extensions are discussed at the end of the paper.

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  1. Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia, Monte S. Angelo, 80126, Naples, Italy

    Francescantonio Oliva

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Oliva, F. Existence and uniqueness of solutions to some singular equations with natural growth. Annali di Matematica 200, 287–314 (2021). https://doi.org/10.1007/s10231-020-00996-1

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