Liouville theorem on a half-space for biharmonic problem with Dirichlet boundary condition

TL;DR

This paper proves Liouville theorems for stable solutions of a nonlinear biharmonic equation with Dirichlet boundary conditions in a half-space, using integral estimates, Pohozaev identity, and monotonicity formulas.

Contribution

It establishes new Liouville type theorems for stable solutions of a biharmonic Hénon type equation in a half-space, extending previous results to this boundary value problem.

Findings

Liouville theorems for stable solutions in half-space

Use of integral estimates and Pohozaev identity

Conditions under which solutions must be trivial

Abstract

We investigate here the nonlinear elliptic H\'enon type equation: $$\D^{2} u= |x|^a|u|^{p-1}u \; \,\,\mbox{in}\,\,\,\, \R^{n}_{+}, \quad \quad u =\frac{\partial u}{\partial x_n} = 0 \quad \mbox{in}\,\,\,\, \partial \R^{n}_{+},$$ with $p>1$ and $n\geq 2$. In particular, we prove some Liouville type theorems for stable at infinity solutions. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.