Positive ground state solutions for a quasilinear elliptic equation with critical exponent

Yinbin Deng; Wentao Huang

doi:10.3934/dcds.2017179

Article Contents

2017, Volume 37, Issue 8: 4213-4230. Doi: 10.3934/dcds.2017179

This issue Previous Article Existence of minimal flows on nonorientable surfaces Next Article Separated nets arising from certain higher rank $\mathbb{R}^k$ actions on homogeneous spaces

Positive ground state solutions for a quasilinear elliptic equation with critical exponent

Yinbin Deng^, and
Wentao Huang

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

^* Corresponding author: Yinbin Deng
^* Corresponding author: Yinbin Deng

Received: October 2016

Revised: March 2017

Published: May 2017

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Abstract

In this paper, we study the following quasilinear elliptic equation with critical Sobolev exponent:

$ -\Delta u +V(x)u-[\Delta(1+u^2)^{\frac 12}]\frac {u}{2(1+u^2)^\frac 12}=|u|^{2^*-2}u+|u|^{p-2}u, \quad x\in {{\mathbb{R}}^{N}}, $

which models the self-channeling of a high-power ultra short laser in matter, where N ≥ 3; 2 < p < 2^* = $\frac{{2N}}{{N -2}}$ and V (x) is a given positive potential. Combining the change of variables and an abstract result developed by Jeanjean in [14], we obtain the existence of positive ground state solutions for the given problem.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J62, 35B33.

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