Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1\left(\Omega \right)$

Betta, M. F. ; Mercaldo, A.  ; Murat, F.  ; Porzio, M. M.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 239-272

Résumé

In this paper we prove uniqueness results for the renormalized solution, if it exists, of a class of non coercive nonlinear problems whose prototype is

$$\left\{\begin{array}{cc}-div\left(a\left(x\right)\right(1+{\left|\nabla u\right|}^2{)}^{\frac{p-2}{2}}{\nabla u)+b\left(x\right)(1+\left|\nabla u\right|}^2{)}^{\frac{\lambda }{2}}=f\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$

where $\Omega$ is a bounded open subset of ${ℝ}^N$, $N\ge 2$, $2-1/N<p<N$, $a$ belongs to $L^{\infty }\left(\Omega \right)$, $a\left(x\right)\ge {\alpha }_0>0$, $f$ is a function in $L^1\left(\Omega \right)$, $b$ is a function in $L^r\left(\Omega \right)$ and $0\le \lambda <{\lambda }^{*}(N,p,r),$ for some $r$ and ${\lambda }^{*}(N,p,r)$.

Détail

DOI : 10.1051/cocv:2002051

Classification : 35J25, 35J60
Keywords: uniqueness, nonlinear elliptic equations, noncoercive problems, data in $L^1$