Concentrating ground state solutions for quasilinear Schrödinger equations with steep potential well

Abstract

We are concerned with the following quasilinear Schrödinger equations (1) $\left\{\begin{array}{l}-\mathrm{\Delta }u+\lambda V\left(x\right)u+\frac{\kappa }{2}\left[\mathrm{\Delta }\right(u^2\left)\right]u=q\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N,\\ u\in H^1\left({\mathbb{R}}^N\right),\end{array}\right.$(1) where $N\ge 3$, $\lambda ,\kappa >0$ are parameters, V and f are nonnegative continuous functions, $q\left(x\right)$ 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.

COMMUNICATED BY:

• S. Leonardi

Keywords:

• Quasilinear Schrödinger equation
• asymptotically linear
• concentrating
• variational methods

2010 Mathematics Subject Classifications:

• 35J20
• 35J25
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by National Postdoctoral Science Foundation of China [grant number 2019M662833] and Natural Science Foundation of China [grant number 11771166].