A class of generalized  quasilinear Schrödinger equations

Yaotian Shen; Youjun  Wang

doi:10.3934/cpaa.2016.15.853

Article Contents

2016, Volume 15, Issue 3: 853-870. Doi: 10.3934/cpaa.2016.15.853

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A class of generalized quasilinear Schrödinger equations

Yaotian Shen^1, and
Youjun Wang^2,

1.
Department of Mathematics, South China University of Technology, Guangzhou 510640
2.
School of Mathematics, South China University of Technology, Guangzhou 510640

Received: May 31, 2015

Revised: October 31, 2015

Published: April 2016

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Abstract

We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with critical Sobolev exponent: \begin{eqnarray} -\Delta u+V(x) u-\Delta [l(u^2)]l'(u^2)u= \lambda u^{\alpha2^*-1}+h(u),\ \ x\in \mathbb{R}^N, \end{eqnarray} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is a given potential and $l,h$ are real functions, $\lambda\geq 0$, $\alpha>1$, $2^*=2N/(N-2)$, $N\geq 3$. Our results cover two physical models $l(s)=s^{\frac{\alpha}{2}}$ and $l(s) = (1+s)^{\frac{\alpha}{2}}$ with $\alpha\geq 3/2$.

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Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35Q55.

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