On a class of periodic quasilinear Schrödinger equations involving critical growth in ${\BR}^2$

Details On a class of periodic quasilinear Schrödinger equations involving critical growth in ${\BR}^2$ On a class of periodic quasilinear Schrödinger equations involving critical growth in ${\BR}^2$

2006-09-08

arxiv.org/abs/math/0609245v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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