Degenerate elliptic equations with nonlinear boundary conditions and measures data

Andreu, Fuensanta; Igbida, Noureddine; Mazón, José M.; Toledo, Julián

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 767-803.

In this paper we study the questions of existence and uniqueness of solutions for equations of type $-\text{div}𝐚(x,Du)+\gamma \left(u\right)\ni {\mu }_1$, posed in an open bounded subset $\Omega$ of ${ℝ}^N$, with nonlinear boundary conditions of the form $𝐚(x,Du)·\eta +\beta \left(u\right)\ni {\mu }_2$. The nonlinear elliptic operator $\text{div}𝐚(x,Du)$ is modeled on the $p$-Laplacian operator ${\Delta }_p\left(u\right)=\text{div}\left({\left|Du\right|}^{p-2}Du\right)$, with $p>1$, $\gamma$ and $\beta$ are maximal monotone graphs in ${ℝ}^2$ such that $0\in \gamma \left(0\right)\cap \beta \left(0\right)$ and the data ${\mu }_1$ and ${\mu }_2$ are measures.

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