Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms

Olivier Guibé; Anna Mercaldo

doi:10.3934/cpaa.2008.7.163

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2008, Volume 7, Issue 1: 163-192. Doi: 10.3934/cpaa.2008.7.163

This issue Previous Article On a class of anisotropic nonlinear elliptic equations in $\mathbb R^N$ Next Article Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation

Uniqueness results for noncoercive nonlinear elliptic equations with two lower order terms

Olivier Guibé^1, and
Anna Mercaldo^2,

1.
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS - Université de Rouen, Avenue de l'Université, BP.12, 76801 Saint-Étienne du Rouvray
2.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universitá di Napoli "Federico II", via Cintia, I-80126 Napoli

Received: September 30, 2006

Revised: March 31, 2007

Published: December 2007

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Abstract

In the present paper we prove uniqueness results for weak solutions to a class of problems whose prototype is

$-d i v((1+|\nabla u|^2)^{(p-2)/2} \nabla u)-d i v(c(x) (1+|u|^2)^{(\tau+1)/2}) $

$+b(x) (1+|\nabla u|^2)^{(\sigma+1)/2}=f \ i n \ \mathcal D'(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

$u\in W^{1,p}_0(\Omega)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

where $\Omega$ is a bounded open subset of $\mathbb R^N$ $(N\ge 2)$, $p$ is a real number $\frac{2N}{N+1}< p <+\infty$, the coefficients $c(x)$ and $b(x)$ belong to suitable Lebesgue spaces, $f$ is an element of the dual space $W^{-1,p'}(\Omega)$ and $\tau$ and $\sigma$ are positive constants which belong to suitable intervals specified in Theorem 2.1, Theorem 2.2 and Theorem 2.3.

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Mathematics Subject Classification: Primary: 35J25; Secondary: 35J60.

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