## On a class of periodic quasilinear Schr\"{o}dinger equations involving critical growth in ${\BR}^2$

### Abstract

We consider the equation $- \Delta u+V(x)u- k(\Del(|u|^{2}))u=g(x,u), u>0, x \in {\BR}^2,$ where $V:{\BR}^2\to {\BR}$ and $g:{\BR}^2 \times {\BR}\to {\BR}$ are two continuous $1-$periodic functions. Also, we assume $g$ behaves like $\exp (\beta |u|^4)$ as $|u|\to \infty.$ We prove the existence of at least one weak solution $u \in H^1({\BR}^2)$ with $u^2 \in H^1({\BR}^2).$ Mountain pass in a suitable Orlicz space together with Moser-Trudinger are employed to establish this result. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schr\"{o}dinger equations. Schr\"{o}dinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics

Topics: Mathematics - Analysis of PDEs, 35J10, 35J20, 35J25
Publisher: 'Elsevier BV'
Year: 2006
DOI identifier: 10.1016/j.jmaa.2007.01.020
OAI identifier: oai:arXiv.org:math/0609245

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