Some Results on the Existence of Solutions for Quasilinear Elliptic Equations with Critical Exponent

Abstract

In this paper, we study the following quasilinear elliptic equation

-\mathrm{\Delta }u+V(x)u-[{\mathrm{\Delta }(1+u^2)}^{\frac{1}{2}}]\frac{u}{2{(1+u^2)}^{\frac{1}{2}}}=f(u)+{|u|}^{2^{\ast }-2}u,x\in {\mathbb{R}}^N,

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\Delta{u}+V(x)u-[{\Delta(1+u^{2})}^{1 \over 2}]{{u} \over 2{(1+u^{2})}^{1 \over 2}}=f(u)+{\vert{u}\vert}^{{2}^{*}-2}u, \quad x \in {\mathbb R}^{N},$$\end{document}

where N ≥ 3, V(x) is a potential function and f satisfies some suitable conditions. We prove the existence of a positive ground state solution via Pohožaev manifold.

Keywords

quasilinear elliptic equation / ground state solution / Pohožaev manifold / 35J15 / 35J20

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Peking University