On the Classification of Stable Solutions of some elliptic equations in half-space

TL;DR

This paper investigates the properties of stable solutions to certain semi-linear elliptic equations in half-space, demonstrating their nonexistence under specific boundary conditions using advanced mathematical techniques.

Contribution

It introduces new nonexistence results for stable solutions of elliptic equations with mixed boundary conditions in half-space, employing Pohozaev identity and monotonicity formulas.

Findings

Proves nonexistence of stable solutions under given conditions

Utilizes Pohozaev identity and monotonicity formulas effectively

Employs blowing down sequence method for analysis

Abstract

In this paper, we are concerned with stable solutions , possibly unbounded and sign-changing, of some semi-linear elliptic problem with mixed nonlinear boundary conditions. We establish the nonexistence of stable solutions, the main methods used are the Pohozaev identity, monotonicity formula of solutions together with a blowing down sequence.