DCDS
Positive ground state solutions for a quasilinear elliptic equation with critical exponent
Yinbin Deng Wentao Huang
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4213-4230 doi: 10.3934/dcds.2017179
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.
keywords: Ground state solutions quasilinear elliptic equation critical exponent
CPAA
Soliton solutions for a quasilinear Schrödinger equation with critical exponent
Wentao Huang Jianlin Xiang
Communications on Pure & Applied Analysis 2016, 15(4): 1309-1333 doi: 10.3934/cpaa.2016.15.1309
This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$ with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort laser in matter. By working with a perturbation approach which was initially proposed in [26], we prove that the given problem has a positive ground state solution.
keywords: Ground state quasilinear Schrödinger equation positive solution perturbation method. critical exponent

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