Existence and asymptotic behavior of ground state solutions for Schrödinger equations with Hardy potential and Berestycki-Lions type conditions

Abstract

In this paper, we investigate the following Schrödinger equation{−Δu−μ|x|2u=g(u)inRN∖{0},u∈H1(RN), where N≥3, μ<(N−2)24, 1|x|2 is called the Hardy potential (the inverse-square potential) and g satisfies the Berestycki-Lions type condition. If 0<μ<(N−2)24, combining variational methods with analytical skills, we show that the above problem has a positive and radial ground state solution. At the same time, our results suggest that this solution together with its derivatives up to order 2 have exponential decay at infinity while this solution has the possibility of blow-up at the origin. Furthermore, we construct a family of ground state solutions which converges to a ground state solution of the limiting problem as μ→0+. If μ<0, we prove that the mountain pass level in H1(RN) can not be achieved. Provided further assumption that the above problem in the radial space Hr1(RN), we obtain the ground state solutions whose energy is strictly greater than the mountain pass level in H1(RN). We also construct a family of solutions which converges to the ground state solution of the limiting problem as μ→0−.