Concentrating ground state solutions for quasilinear Schrödinger equations with steep potential well. Issue 14 (26th October 2021) - Jia, Huifang
Abstract : We are concerned with the following quasilinear Schrödinger equations (1) − Δ u + λ V ( x ) u + κ 2 [ Δ ( u 2 ) ] u = q ( x ) f ( u ),   x ∈ R N, u ∈ H 1 ( R N ), where N ≥ 3, λ, κ > 0 are parameters, V and f are nonnegative continuous functions, q ( x ) is a positive bounded function. By using variational methods, we study the existence of positive ground state solutions to problem (1) when V,   q and f satisfy some suitable conditions. Furthermore, the concentrating behavior of ground state solutions to problem (1) is proved. We mainly extend the results in Severo, Gloss and da Silva [On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms. J Differ Equ. 2017;263:3550–3580], which considered quasilinear Schrödinger equations with positive potential function, to quasilinear Schrödinger equations with steep potential well.