Liouville type theorems for two mixed boundary value problems with general nonlinearities

In this paper, we study the nonexistences of positive solutions for the following two mixed boundary value problems. The first problem is the nonlinear-Neumann mixed boundary value problem{−Δu=f(u)inR+N,∂u∂ν=g(u)onΓ1,∂u∂ν=0onΓ0 and the second is the nonlinear-Dirichlet mixed boundary value problem{−Δu=f(u)inR+N,∂u∂ν=g(u)onΓ1,u=0onΓ0, where N≥3, R+N={x∈RN:xN>0}, Γ1={x∈RN:xN=0,x1<0} and Γ0={x∈RN:xN=0,x1>0}. We will prove that these problems possess no positive solution under some assumptions on the nonlinear terms. The main method we use is the moving plane method in an integral form.