An elliptic problem withdegenerate coercivityand a singular quadratic gradient lower order term

Gisella Croce

doi:10.3934/dcdss.2012.5.507

Article Contents

2012, Volume 5, Issue 3: 507-530. Doi: 10.3934/dcdss.2012.5.507

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An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term

Gisella Croce^1,

1.
Laboratoire de Mathématiques Appliquées du Havre, Université du Havre, 25, rue Philippe Lebon, 76063 Le Havre

Received: July 31, 2010

Revised: August 31, 2010

Published: May 2012

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Abstract

In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. The model problem is $$ \begin{equation} \left\{\begin{array}{11} -div\left(\frac{\nabla u}{(1+|u|)^p}\right) + \frac{|\nabla u|^{2}}{|u|^{\theta}} = f & \mbox{in $\Omega$,} \\ u = 0 & \mbox{on $\partial\Omega$,} \end{array} \right. \end{equation} $$ where $\Omega$ is an open bounded set of $\mathbb{R}^N$, $N\geq 3$ and $p, \theta>0$. The source $f$ is a positive function belonging to some Lebesgue space. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions, when $p>\theta-1$ and $\theta<2$.

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Mathematics Subject Classification: 35J15, 35J25, 35J66, 35J70, 35J75.

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References

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