Elliptic equations having a singular quadratic gradient term anda changing sign datum

Daniela Giachetti; Francesco  Petitta; Sergio Segura de León

doi:10.3934/cpaa.2012.11.1875

Article Contents

2012, Volume 11, Issue 5: 1875-1895. Doi: 10.3934/cpaa.2012.11.1875

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Elliptic equations having a singular quadratic gradient term and a changing sign datum

1.
Dip. Metodi e Modelli Matematici per le Scienze Applicate, Univ. Roma 1, Via Antonio Scarpa 16, 00161 Roma
2.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza, Università di Roma, Via Scarpa 16, 00161 Roma
3.
Departament d'An

Received: March 2011

Revised: February 2012

Published: September 2012

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Abstract

In this paper we study a singular elliptic problem whose model is \begin{eqnarray*} - \Delta u= \frac{|\nabla u|^2}{|u|^\theta}+f(x), in \Omega\\ u = 0, on \partial \Omega; \end{eqnarray*} where $\theta\in (0,1)$ and $f \in L^m (\Omega)$, with $m\geq \frac{N}{2}$. We do not assume any sign condition on the lower order term, nor assume the datum $f$ has a constant sign. We carefully define the meaning of solution to this problem giving sense to the gradient term where $u=0$, and prove the existence of such a solution. We also discuss related questions as the existence of solutions when the datum $f$ is less regular or the boundedness of the solutions when the datum $f \in L^m (\Omega)$ with $m> \frac{N}{2}$.

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Mathematics Subject Classification: 35J75, 35J25.

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