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Uniqueness for Neumann problems for nonlinear elliptic equations

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  • In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is

    $\left\{ \begin{align} & -\text{div}({{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u)-\text{div}(c(x)|u{{|}^{p-2}}u)=f\ \ \ \text{in}\ \Omega , \\ & \left( {{(1+|\nabla u{{|}^{2}})}^{(p-2)/2}}\nabla u+c(x)|u{{|}^{p-2}}u \right)\cdot \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{n}=0\ \ \ \text{on}\ \partial \Omega , \\ \end{align} \right.$

    where $Ω$ is a bounded domain of $\mathbb{R}^{N}$, $N≥ 2$, with Lipschitz boundary, $ 1 < p < N$, $\underline n$ is the outer unit normal to $\partial Ω$, the datum $f$ belongs to $L^{(p^{*})'}(Ω)$ or to $L^{1}(Ω)$ and satisfies the compatibility condition $\int{{}}_Ω f \, dx = 0$. Finally the coefficient $c(x)$ belongs to an appropriate Lebesgue space.

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


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