Existence and asymptotic behavior of solutions for a quasilinear Schrödinger equation with Hardy potential

Abstract

In this paper, we discuss the quasilinear Schrödinger equation with a Hardy potential (Pμ)−Δu−μu|x|2−Δ(u2)u=g(u),x∈RN∖{0},u∈H1(RN),where N≥3, μ<μ̄=(N−2)24, 1|x|2 is called the Hardy potential and g∈C(R,R) satisfies the Berestycki–Lions type conditions. When 0<μ<(N−2)24, we show that the above problem has a positive and radial solution ū∈H1(RN). At the same time, we prove that the solution ū together with its derivatives up to order 2 have exponential decay at infinity and the solution has the possibility of blow-up at the origin. When μ<0, we obtain a solution ũ of (Pμ) in the radial space Hr1(RN) and we also prove that the solution ũ together with its derivatives up to order 2 have exponential decay at infinity and the solution has the possibility of blow-up at the origin. Furthermore, we construct a family of solutions which converge to a solution of the limiting problem as μ→0.