PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Volume 116 / 2016

Lin Chen, Caisheng Chen, Zonghu Xiu 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.

Authors

  • Lin ChenCollege of Science
    Hohai University
    210098 Nanjing, P.R. China
    and
    College of Mathematics and Statistics
    Yili Normal University
    835000 Yining, P.R. China
    e-mail
  • Caisheng ChenCollege of Science
    Hohai University
    210098 Nanjing, P.R. China
    e-mail
  • Zonghu XiuScience and Information College
    Qingdao Agricultural University
    266109 Qingdao, P.R. China
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image