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

In this paper, we discuss the quasilinear Schrodinger equation with a Hardy potential {-.u - mu u |x|2 -.(u2) u = g(u), x. RN \ {0}, u. H1(RN), ( P mu) where N = 3, mu < <(mu)over bar> = ( N-2)2 4, 1 |x|2 is called the Hardy potential and g. C(R, R) satisfies the Berestycki-Lions type conditions. When 0 < mu < ( N-2)2 4, we show that the above problem has a positive and radial solution (u) over bar. H1(RN). At the same time, we prove that the solution (u) over bar 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 mu < 0, we obtain a solution u of (P mu) in the radial space H1 r (RN) and we also prove that the solution u 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 mu. 0. (C) 2022 Published by Elsevier Ltd.