Liouville Type Theorems for Two Mixed Boundary Value Problems with General Nonlinearities

Abstract

In this paper, we study the nonexistence of positive solutions for the following two mixed boundary value problems. The first problem is the mixed nonlinear-Neumann boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ \frac{\partial u}{\partial \nu}=0 &{\rm on}\quad \Gamma_0 \end{array} \right. $$ and the second is the nonlinear-Dirichlet boundary value problem $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle \\ u=0 &{\rm on}\quad \Gamma_0, \end{array} \right. $$ where $\R=\{x\in \mathbb R^N:x_N>0\}$, $\Gamma_1=\{x\in \mathbb R^N:x_N=0,x_1<0\}$ and $\Gamma_0=\{x\in \mathbb R^N:x_N=0,x_1>0\}$. We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms. The main technique we use is the moving plane method in an integral form.