The “Blow up” problem for a quasilinear Schrödinger equation

Abstract

Consider the blow up results for W2,2(RN) solutions of a quasilinear Schrodinger equation iut+Δu+β∣u∣p−2u+θ(Δ∣u∣2)u=0, u∣t=0=u0(x),x∊RN. When ∣x∣u0∊L2(RN), we show that the W2,2(RN) solutions must blow up for any β⩾0, θ∊R and some restriction on p. We also show that the radial symmetric solutions in W2,2(RN) must blow up at finite time without assuming ∣x∣u0∊L2(RN).