# Nonlinear elliptic equations on the upper half space

## TL;DR

This paper classifies all positive solutions to a class of nonlinear elliptic equations with boundary conditions on the upper half space, revealing their dependence on variables and symmetry properties based on parameter ranges.

## Contribution

It provides a complete classification of positive solutions for specific nonlinear elliptic equations with boundary conditions, including symmetry and variable dependence results.

## Key findings

- Solutions depend only on the last variable for certain parameters.
- Solutions are either variable-dependent or radially symmetric at critical parameter values.
- The classification covers all positive solutions under the specified conditions.

## Abstract

In this paper we shall classify all positive solutions of $ \Delta u =a u^p$ on the upper half space $ H =\Bbb{R}_+^n$ with nonlinear boundary condition $ {\partial u}/{\partial t}= - b u^q $ on $\partial H$ for both positive parameters $a, \ b>0$. We will prove that for $p \ge {(n+2)}/{(n-2)}, 1\leq q<{n}/{(n-2)}$ (and $n \ge 3$) all positive solutions are functions of last variable; for $p= {(n+2)}/{(n-2)}, q= {n}/{(n-2)}$ (and $n \ge 3$) positive solutions must be either some functions depending only on last variable, or radially symmetric functions.