In this paper we present a unified approach toinvestigate existence and concentration of positive solutions for the following class of quasilinear Schrödingerequations,-ε2Δu+V(x)u∓ε2+γuΔu2=h(u),x∈RN,where N⩾ 3 , ε> 0 , V(x) is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter γ> 0. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution uε,γ concentrating, as ε→ 0 , around minima points of the potential.

A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations

Cassani D.;Wang Y.;Zhang J.
2020

Abstract

In this paper we present a unified approach toinvestigate existence and concentration of positive solutions for the following class of quasilinear Schrödingerequations,-ε2Δu+V(x)u∓ε2+γuΔu2=h(u),x∈RN,where N⩾ 3 , ε> 0 , V(x) is a positive external potential,h is a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameter γ> 0. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solution uε,γ concentrating, as ε→ 0 , around minima points of the potential.
critical growth; quasilinear Schrödinger equations; Semiclassical states; variational methods
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11383/2101534
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