Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in ${L}^{1}\left(\Omega \right)$
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 239-272.

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)\left(1+{|\nabla u|}^{2}{\right)}^{\frac{p-2}{2}}{\nabla u\right)+b\left(x\right)\left(1+|\nabla u|}^{2}{\right)}^{\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, $a$ 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 $0\le \lambda <{\lambda }^{*}\left(N,p,r\right),$ for some $r$ and ${\lambda }^{*}\left(N,p,r\right)$.

DOI: 10.1051/cocv:2002051
Classification: 35J25,  35J60
Keywords: uniqueness, nonlinear elliptic equations, noncoercive problems, data in ${L}^{1}$
@article{COCV_2002__8__239_0,
author = {Betta, M. F. and Mercaldo, A. and Murat, F. and Porzio, M. M.},
title = {Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {239--272},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
doi = {10.1051/cocv:2002051},
zbl = {1092.35032},
mrnumber = {1932952},
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Betta, M. F.; Mercaldo, A.; Murat, F.; Porzio, M. M. Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1(\Omega )$. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 239-272. doi : 10.1051/cocv:2002051. http://archive.numdam.org/articles/10.1051/cocv:2002051/

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