P-Harmonic Functions in ℝ N + with Nonlinear Neumann Boundary Conditions and Measure Data

Abstract

Abstract We study a concept of renormalized solution to the problem { - Δ p ⁢ u = 0 in ⁢ ℝ + N , | ∇ ⁡ u | p - 2 ⁢ u ν + g ⁢ ( u ) = μ on ⁢ ∂ ⁡ ℝ + N , \begin{cases}-\Delta_{p}u=0&\mbox{in }{\mathbb{R}}^{N}_{+},\\ \lvert\nabla u\rvert^{p-2}u_{\nu}+g(u)=\mu&\mbox{on }\partial{\mathbb{R}}^{N}_% {+},\end{cases} where 1 < p ≤ N {1<p\leq N} , N ≥ 2 {N\geq 2} , ℝ + N = { ( x ′ , x N ) : x ′ ∈ ℝ N - 1 , x N > 0 } {{\mathbb{R}}^{N}_{+}=\{(x^{\prime},x_{N}):x^{\prime}\in{\mathbb{R}}^{N-1},\,x% _{N}>0\}} , u ν {u_{\nu}} is the normal derivative of u, μ is a bounded Radon measure, and g : ℝ → ℝ {g:{\mathbb{R}}\rightarrow{\mathbb{R}}} is a continuous function. We prove 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 absorption type in both the subcritical and supercritical case. For the problem with source we study the power nonlinearity g ⁢ ( u ) = - u q {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 μ ≡ 0 {\mu\equiv 0} and g ⁢ ( u ) = - u q {g(u)=-u^{q}} in the supercritical case.