Existence of solutions for a resonant Steklov Problem - doi: 10.5269/bspm.v27i1.9070

Abstract

In this paper, we prove the existence of weak solutions to the problem \Delta_p u = 0 in \Omega and |∇u|^{p−2}\frac{\partial u} {\partial \nu}= \lambda_1 m(x)|u|^{p−2}u + f(x, u) − h on \partial \Omega, where \Omega is a bounded domain in R^N (N ≥ 2), m ∈ L^q(\partial \Omega) is a weight, \lambda_1 is the first positive eigenvalue for the eigenvalue Steklov problem \Delta_pu = 0 in  \Omega and |∇u|^{p−2} \frac{\partial u} {\partial \nu} =\lambda m(x)|u|^{p−2}u on \partial \Omega. f and h are functions that satisfy some conditions.