Quasilinear Schrödinger equations involving concave and convex nonlinearities

João Marcos Do Ó,Uberlandio Severo

doi:10.3934/cpaa.2009.8.621

Abstract

In this paper the mountain--pass theorem and the Ekeland variationalprinciple in a suitable Orlicz space are employed to establish theexistence of positive standing wave solutions for a quasilinearSchrödinger equation involving a combination of concave andconvex terms. The second order nonlinearity considered in this papercorresponds to the superfluid equation in plasma physics.

Highlights

In this paper we are concerned with quasilinear Schrodinger equations of the form ∂ψ i= −∆ψ + W (x)ψ − η(|ψ|2)ψ − κ ∆ρ(|ψ|2) ρ (|ψ|2)ψ, (1)∂t where ψ : R × RN → C, κ is a positive constant, W : RN → R is a given potential and ρ, η : R+ → R are suitable functions
In this paper the mountain–pass theorem and the Ekeland variational principle in a suitable Orlicz space are employed to establish the existence of positive standing wave solutions for a quasilinear Schrodinger equation involving a combination of concave and convex terms
The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics

Summary

Introduction

In this paper we are concerned with quasilinear Schrodinger equations of the form ∂ψ i= −∆ψ + W (x)ψ − η(|ψ|2)ψ − κ ∆ρ(|ψ|2) ρ (|ψ|2)ψ, (1)∂t where ψ : R × RN → C, κ is a positive constant, W : RN → R is a given potential and ρ, η : R+ → R are suitable functions. QUASILINEAR SCHRO DINGER EQUATIONS INVOLVING CONCAVE AND CONVEX NONLINEARITIES Departamento de Matematica Universidade Federal da Paraıba 58051-900, Joao Pessoa – PB – Brazil (Communicated by Zhi-Qiang Wang)

Results

Conclusion