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### 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

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 ${ℝ}^{N}$, $N\ge 2$, $2-1/N<p<N$, $a$ 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 $0\le \lambda <{\lambda }^{*}\left(N,p,r\right),$ for some $r$ and ${\lambda }^{*}\left(N,p,r\right)$.

### Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in  (Ω)

ESAIM: Control, Optimisation and Calculus of Variations

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\{-\text{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\text{in}\phantom{\rule{1.0em}{0ex}}\Omega ,u=0\text{on}\phantom{\rule{1.0em}{0ex}}\partial \Omega ,\right\$ where Ω is a bounded open subset of ${ℝ}^{N}$, N > 2, 2-1/, belongs to  (Ω), $a\left(x\right)\ge {\alpha }_{0}>0$, is a function in (Ω), is a function in ${L}^{r}\left(\Omega \right)$ and 0 ≤ λ < λ *(), for some and λ *().

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