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

Original · EN

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 { arrayll -Δu=f(u) & in, ∂ u/∂ ν=g(u) & on Γ₁, ∂ u/∂ ν=0 & on Γ₀ array. and the second is the nonlinear-Dirichlet boundary value problem { arrayll -Δu=f(u) & in, ∂ u/∂ ν=g(u) & on Γ₁, u=0 & on Γ₀, array. where ={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 technique we use is the moving plane method in an integral form.