Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential

Boumediene Abdellaoui; Daniela Giachetti; Ireneo Peral; Magdalena Walias

doi:10.3934/dcds.2014.34.1747

Article Contents

2014, Volume 34, Issue 5: 1747-1774. Doi: 10.3934/dcds.2014.34.1747

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Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential

1.
Laboratoire D'Analyse Nonlinéaire et Mathématiques Appliquées, Université Aboubekr Belkaïd, Tlemcen, Tlemcen 13000
2.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Università di Roma Sapienza, via Scarpa 16, 00161 Roma
3.
Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid
4.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid

Received: June 30, 2013

Revised: July 31, 2013

Published: April 2014

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Abstract

In this article we consider the following family of nonlinear elliptic problems,
$-\Delta (u^m) - \lambda \frac{u^m}{|x|^2} = |Du|^q + c f(x). $
We will analyze the interaction between the Hardy-Leray potential and the gradient term getting existence and nonexistence results in bounded domains $\Omega\subset\mathbb{R}^N$, $N\ge 3$, containing the pole of the potential.
Recall that $Λ_N = (\frac{N-2}{2})^2$ is the optimal constant in the Hardy-Leray inequality.
1.For $0 < m \le 2$ we prove the existence of a critical exponent $q_+ \le 2$ such that for $q > q_+$, the above equation has no positive distributional solution. If $q < q_+$ we find solutions by using different alternative arguments.
Moreover if $q = q_+ > 1$ we get the following alternative results.
(a) If $m < 2$ and $q=q_+$ there is no solution.
(b) If $m = 2$, then $q_+=2$ for all $\lambda$. We prove that there exists solution if and only if $2\lambda\leq\Lambda_N$ and, moreover, we find infinitely many positive solutions.
2. If $m > 2$ we obtain some partial results on existence and nonexistence.
We emphasize that if $q(\frac{1}{m}-1)<-1$ and $1 < q \le 2$, there exists positive solutions for any $f \in L^1(Ω)$.

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Mathematics Subject Classification: 35A01, 35D30, 35J25, 35J70, 35J60, 35J75.

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