$p$-harmonic functions in $\mathbb{R}^N_+$ with nonlinear Neumann boundary conditions and measure data

Abstract

We propose and study a concept of renormalized solution to the problem $\Delta_p u=0$ in $\mathbb{R}^N_+$, $|\nabla u|^{p-2}u_{\nu} + g(u) = \mu$ on $\partial\mathbb{R}^N_+$, where $1<p\leq N$, $N\geq 2$, $\mathbb{R}^N_+=\left\lbrace(x',x_N):x'\in\mathbb{R}^{N-1}, x_N>0\right\rbrace $, $u_{\nu}$ is the normal derivative of $u$, $\mu$ is a bounded Radon measure, and $g:\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear term. We develop stability results and, using the symmetry of the domain, apriori estimates on hyperplanes, and potential methods, we obtain several existence results. In particular, we show existence of solutions for problems with nonlinear terms of the absorption type in both subcritical and supercritical cases. Regarding the problem with source, we study the power nonlinearity $g(u)=-u^q$, showing existence in the supercritical case, and nonexistence in the subcritical one. We also give a characterization of removable sets when $\mu\equiv 0$ and $g(u)=-u^q$ in the supercritical case. We remark that this work is motivated by similar results obtained for the problem $-\Delta_p u + g(x,u)=\mu$ in bounded domains.