A nonlinear elliptic problem involving the gradient on a half space

TL;DR

This paper proves the existence of classical solutions to a perturbed nonlinear elliptic PDE involving the gradient on a half space, for certain ranges of the exponent p and decay conditions on the perturbation g.

Contribution

It establishes existence results for a class of nonlinear elliptic problems with gradient dependence and perturbations, extending previous understanding to specific p ranges and decay conditions.

Findings

Existence of classical solutions for p in (4/3, 2)

Solutions exist under bounded perturbations with uniform radial decay

The problem extends the theory of nonlinear elliptic equations involving gradients

Abstract

We consider perturbations of the diffusive Hamilton-Jacobi equation \begin{equation*} %\label{non_pert} \left\{ \begin{array}{lcl} \hfill -\Delta u &=& (1+g(x))| \nabla u|^p\qquad \mbox{ in } \IR^N_+, \\ \hfill u &=& 0 \hfill \mbox{ on } \partial \IR^N_+, \end{array}\right. \end{equation*} for $ p>1$. We prove the existence of a classical solution provided $ p \in (\frac{4}{3},2)$ and $g$ is bounded with uniform radial decay to zero.