A quasilinear singular elliptic problem related to the Kardar–Parisi–Zhang equation

ABSTRACT

In this paper, we will study existence and properties of solutions for the nonlinear Dirichlet problem $\left\{\begin{array}{ll}-\mathrm{div}\left(M\right(x\left)\mathrm{\nabla }u\right)=b\left(x\right)\frac{|\mathrm{\nabla }u|}{u^{\theta }}+f\left(x\right),& \text{in }\mathrm{\Omega },\\ u=0,& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}\right.$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain, $0<\theta \le 1,$ M is a uniformly elliptic and bounded matrix, and b, f are functions in some Lebesgue spaces.

Keywords:

• Kardar–Parisi–Zhang equation
• Quasilinear singular PDE

2010 Mathematics Subject Classifications:

• 35J62
• 35B65
• 35J75

Acknowledgements

Part of this work has been carried out during the stays of the first author at the Departamento de Matemàticas of Universidad Autónoma de Madrid and at the ‘Dipartimento di chirurgia generale e specialistica’ of Sapienza for a cooperation with Professor Massimo Rossi. We also thank the referee for the useful remarks on the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is done with the support of the projects MTM2013-40846-P and MTM2016-80474-P, Spain.