Existence and asymptotic behavior of solutions for a quasilinear Schrödinger equation with Hardy potential. (November 2022) - Hu, Die; Tang, Xianhua; Zhang, Qi
Abstract: In this paper, we discuss the quasilinear Schrödinger equation with a Hardy potential ( P μ ) − Δ u − μ u | x | 2 − Δ ( u 2 ) u = g ( u ), x ∈ R N ∖ { 0 }, u ∈ H 1 ( R N ), where N ≥ 3, μ < μ ̄ = ( 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 < μ < ( N − 2 ) 2 4, we show that the above problem has a positive and radial solution u ̄ ∈ H 1 ( R N ) . At the same time, we 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. When μ < 0, we obtain a solution u ̃ of ( P μ ) in the radial space H r 1 ( R N ) 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 μ → 0 .