Multi-bump bound state solutions for the quasilinear Schrödinger equation with critical frequency

Yuxia Guo and Zhongwei Tang

Vol. 270 (2014), No. 1, 49–77

DOI: 10.2140/pjm.2014.270.49

Abstract

We study the existence of single- and multi-bump solutions of quasilinear Schrödinger equations

$$-\Delta u+\lambda V\left(x\right)u-\frac{1}{2}\left(\Delta |u{|}^2\right)u=|u{|}^{p-2}u,$$

the function $V$ being a critical frequency in the sense that $\underset{x\in {ℝ}^N}{inf}V\left(x\right)=0$. We show that if the zero set of $V$ has several isolated connected components ${\Omega }_1,\dots ,{\Omega }_k$ such that the interior of ${\Omega }_i$ is not empty and $\partial {\Omega }_i$ is smooth, then for $\lambda >0$ large, there exists, for any nonempty subset $J\subset \left\{1,2,\dots ,k\right\}$, a standing wave solution trapped in a neighborhood of $\bigcup _{j\in J}{\Omega }_j$.

Keywords

multi-bump bound states, quasilinear Schrödinger equation, Orlicz space

Mathematical Subject Classification 2010

Primary: 35Q55

Secondary: 35J65

Milestones

Received: 16 December 2012

Revised: 19 February 2013

Accepted: 27 September 2013

Published: 2 August 2014