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

Authors: Shilpa Gupta ; Gaurav Dwivedi

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.