Communications on Pure and Applied Analysis (CPAA)

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

Pages: 163 - 192, Volume 7, Issue 1, January 2008      doi:10.3934/cpaa.2008.7.163

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Olivier Guibé - Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS - Université de Rouen, Avenue de l'Université, BP.12, 76801 Saint-Étienne du Rouvray, France (email)
Anna Mercaldo - Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universitá di Napoli "Federico II", via Cintia, I-80126 Napoli, Italy (email)

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.

Keywords:  Uniqueness, nonlinear elliptic equations, noncoercive problems, weak solutions.
Mathematics Subject Classification:  Primary: 35J25; Secondary: 35J60.

Received: October 2006;      Revised: April 2007;      Available Online: October 2007.