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.