The elliptic anisotropic problem with Wentzell boundary conditions and variable exponents

Abstract

Let $\Omega\subseteq\mathbb{R\!}^N$ be a bounded Lipschitz domain, for $N\geq3$. We investigate the solvability and regularity of a class of quasi-linear elliptic equations involving the anisotropic $\overset{\rightarrow}p(\cdot)$-Laplace operator $\Delta_{\overset{\rightarrow}p(\cdot)}$ with nonhomogeneous anisotropic Wentzell boundary conditions $$ \displaystyle\sum^N_{i=1} \left|\frac{\partial u}{\partial x_i}\right|^{p_i(\cdot)-2}\displaystyle\frac{\partial u}{\partial x_i}\nu_i- \Delta_{_{\overset{\rightarrow}q(\cdot),\Gamma}}u+ \beta|u|^{q_M(\cdot)-2}u\,=\,g\,\,\,\,\,\textrm{on}\,\,\,\Gamma:=\partial\Omega, $$ for $\beta\in L^{\infty}(\Gamma)^+$ with a positive essential lower bound, where $\Delta_{_{\overset{\rightarrow}q(\cdot),\Gamma}}$ denotes the anisotropic $\overset{\rightarrow}q(\cdot)$-Laplace-Beltrami operator, and $q_M(x)=\max\{q_1(x), \ldots,q_{N-1}(x)\}$. Under minimal conditions, we establish existence and uniqueness of weak solutions for the elliptic problem, and moreover, we prove that such solutions are globally bounded over $\overline{\Omega}$. Inverse positivity and comparison results are also obtained. At the end, we establish a nonlinear Fredholm alternative for the anisotropic Wentzell problem.