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

TL;DR

This paper establishes Liouville type theorems demonstrating the nonexistence of positive solutions for certain mixed boundary value problems with nonlinearities in half-space domains, using the moving plane method in integral form.

Contribution

It introduces new nonexistence results for mixed boundary problems with general nonlinearities, extending Liouville theorems to these boundary conditions.

Findings

No positive solutions exist under specified nonlinear conditions.

The moving plane method in integral form is effective for these problems.

Results apply to both nonlinear-Neumann and nonlinear-Dirichlet boundary conditions.

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.