Robin problems for the \(p\)-Laplacian with gradient dependence
Let \(\Omega \subset \mathbb{R}^N\) be a bounded smooth domain. The paper is concerned with the following \(p\)-Laplacian type Robin boundary problem: \[ -\operatorname{div}(|Du|^{p-2}Du(z)) = f(z, u(z),Du(z)) \text{ in } \Omega, \ \ \frac{\partial u}{{\partial n}_p} + \beta(z)|u|^{p-2}u = 0 \text{ on } \partial \Omega,\tag{P} \] where \(p\in (0,\infty)\), \(f(z, x, y)\) is a Carathéodory function, \(\frac{\partial u}{{\partial n}_p}\) denotes the conormal derivative and \(\beta(z)\geq 0\) is Hölder continuous on \(\partial \Omega\). Under some further conditions on \(f(z, x, y)\), such as \(f(z, x, y) \leq \Theta (z) |x|^{p-2}x\) uniformly for a.e. \(z\in \Omega\) and all \( y\in \mathbb{R}^N\), as \(|x| \rightarrow \infty\), if \( \Theta(z)\) is bounded by certain constant, the authors prove the existence of a smooth solution for problem (P) by the Leray-Schauder alternative principle.
scientific article; zbMATH DE number 7043507