Positive solution for a quasilinear equation with critical growth in $\mathbb {R}^N$

Lin Chen¹ ; Caisheng Chen² ; Zonghu Xiu³

¹ College of Science Hohai University 210098 Nanjing, P.R. China and College of Mathematics and Statistics Yili Normal University 835000 Yining, P.R. China
² College of Science Hohai University 210098 Nanjing, P.R. China
³ Science and Information College Qingdao Agricultural University 266109 Qingdao, P.R. China

Annales Polonici Mathematici, Tome 116 (2016) no. 3, pp. 251-262

Résumé

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.

Détail

DOI : 10.4064/ap3664-1-2016

Keywords: study existence positive solutions quasilinear problem begin equation* begin array vardelta x n nabla n nabla quad mathbb quad mathbb end array right end equation* where vardelta mathop div nolimits nabla n nabla n laplacian operator mathbb rightarrow mathbb continuous potential mathbb times mathbb rightarrow mathbb continuous function main result follows iterative method based mountain pass techniques