We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -\Delta u = f(x) \text{ in }\Omega, \frac{\partial u}{\partial \nu} + \lambda(x) u = \frac{g(x)}{u^\eta} \text{ on } \partial\Omega, \end{array} \right. \end{equation*} where $\eta\ge 0$ and $f,\lambda$ and $g$ are nonnegative integrable functions. The set $\Omega\subset\mathbb{R}^N (N> 2)$ is open and bounded with smooth boundary and $\nu$ denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of $p$-Laplacian type jointly with nonlinear boundary conditions. We prove existence of an entropy solution and check that this solution is unique under natural assumptions. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.
On a nonlinear Robin problem with an absorption term on the boundary and $L^1$ data / DELLA PIETRA, Francesco; Oliva, Francescantonio; Segura de León, Sergio. - In: ADVANCES IN NONLINEAR ANALYSIS. - ISSN 2191-950X. - 13:(2024), pp. 1-16. [10.1515/anona-2023-0118]
On a nonlinear Robin problem with an absorption term on the boundary and $L^1$ data
Francesco Della Pietra
;
2024
Abstract
We deal with existence and uniqueness of nonnegative solutions to \begin{equation*} \left\{ \begin{array}{l} -\Delta u = f(x) \text{ in }\Omega, \frac{\partial u}{\partial \nu} + \lambda(x) u = \frac{g(x)}{u^\eta} \text{ on } \partial\Omega, \end{array} \right. \end{equation*} where $\eta\ge 0$ and $f,\lambda$ and $g$ are nonnegative integrable functions. The set $\Omega\subset\mathbb{R}^N (N> 2)$ is open and bounded with smooth boundary and $\nu$ denotes its unit outward normal vector. More generally, we handle equations driven by monotone operators of $p$-Laplacian type jointly with nonlinear boundary conditions. We prove existence of an entropy solution and check that this solution is unique under natural assumptions. Among other features, we study the regularizing effect given to the solution by both the absorption and the nonlinear boundary term.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.