Positive solution for a quasilinear equation with critical growth in $\mathbb {R}^N$
Volume 116 / 2016
Annales Polonici Mathematici 116 (2016), 251-262
MSC: Primary 35J20; Secondary 35J62.
DOI: 10.4064/ap3664-1-2016
Published online: 4 January 2016
Abstract
We study the existence of positive solutions of the quasilinear problem \begin{equation*} \left\{ \begin{array}{@{}l@{}} -\varDelta_N u+V(x)|u|^{N-2}u=f(u,|\nabla u|^{N-2}\nabla u),\quad x\in \mathbb{R}^N,\\ u(x) \gt 0,\quad x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $ \varDelta_N u =\mathop{\rm div}\nolimits (|\nabla u|^{N-2}\nabla u)$ is the $N$-Laplacian operator, $V:\mathbb{R}^N \rightarrow \mathbb{R}$ is a continuous potential, $f:\mathbb{R}\times \mathbb{R}^N \rightarrow \mathbb{R}$ is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.
