An existence result for $p$-Laplace equation with gradient nonlinearity in $\mathbb{R}^N$

TL;DR

This paper establishes the existence of positive weak solutions for a $p$-Laplace equation with gradient-dependent nonlinearity in $ ^N$, using variational methods and the Mountain pass theorem.

Contribution

It provides the first existence proof for solutions to a $p$-Laplace problem with gradient nonlinearity in unbounded domains, employing an iterative variational approach.

Findings

Existence of positive weak solutions proved.

Application of Mountain pass theorem to nonlinear PDE.

Solution existence established for $1<p<N$.

Abstract

We prove the existence of a weak solution to the problem \begin{equation*} \begin{split} -\Delta_{p}u+V(x)|u|^{p-2}u & =f(u,|\nabla u|^{p-2}\nabla u), \ \ \ \\ u(x) & >0\ \ \forall x\in\mathbb{R}^{N}, \end{split} \end{equation*} where $\Delta_{p}u=\hbox{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace operator, $1<p<N$ and the nonlinearity $f:\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ is continuous and it depends on gradient of the solution. We use an iterative technique based on the Mountain pass theorem to prove our existence result.