## Shilpa Gupta ; Gaurav Dwivedi - An existence result for $p$-Laplace equation with gradient nonlinearity in $\mathbb{R}^N$

cm:9316 - Communications in Mathematics, May 23, 2022, Volume 30 (2022), Issue 1 - https://doi.org/10.46298/cm.9316
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.

Volume: Volume 30 (2022), Issue 1
Published on: May 23, 2022
Imported on: July 17, 2020
Keywords: Mathematics - Analysis of PDEs,35J20, 35J62, 35J92

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