Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations

Daniele Cassani; João Marcos do Ó; Abbas Moameni

doi:10.3934/cpaa.2010.9.281

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2010, Volume 9, Issue 2: 281-306. Doi: 10.3934/cpaa.2010.9.281

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Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations

1.
Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano
2.
Departamento de Matemática, Universidade Fededral da Paraíba, 58059-900, João Pessoa-PB
3.
Department of Mathematics and Statistics, Queen’s University Jeffery Hall, University Ave. Kingston, ON Canada, K7L 3N6

Received: December 31, 2008

Revised: July 31, 2009

Published: February 2010

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Abstract

In this paper we prove the existence and qualitative properties of positive bound state solutions for a class of quasilinear Schrödinger equations in dimension $N\ge 3$: we investigate the case of unbounded potentials which can not be covered in a standard function space setting. This leads us to use a nonstandard penalization technique, in a borderline Orlicz space framework, which enable us to build up a one parameter family of classical solutions which have finite energy and exhibit, as the parameter goes to zero, a concentrating behavior around some point which is localized.

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Mathematics Subject Classification: 35J10; 35J20; 35J25.

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