Ground state solutions for a class of quasilinear Schrödinger equations with Choquard type nonlinearity

Abstract In this paper, we study the following Choquard type quasilinear Schrodinger equation − Δ u + V ( x ) u − Δ ( u 2 ) u = ( I α ∗ | u | p ) | u | p − 2 u , x ∈ R N , where N ≥ 3 , 0 α N , 2 ( N + α ) N p 2 ( N + α ) N − 2 , V : R N → R is radial potential and I α is a Riesz potential. Using the method developed by Jeanjean (1999), we establish the existence of ground state solutions.

[1]  Louis Jeanjean,et al.  On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  Shaoxiong Chen,et al.  Existence of positive solutions for a class of quasilinear Schrödinger equations of Choquard type , 2019, Journal of Mathematical Analysis and Applications.

[3]  Jiaquan Liu,et al.  Quasilinear elliptic equations via perturbation method , 2012 .

[4]  Andrzej Szulkin,et al.  Multiple solutions for a quasilinear Schrödinger equation , 2013 .

[5]  Wei Zhang,et al.  Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method , 2018, Journal of Mathematical Physics.

[6]  Jiaquan Liu,et al.  Soliton solutions for quasilinear Schrödinger equations, II , 2003 .

[7]  J. García Azorero,et al.  Hardy Inequalities and Some Critical Elliptic and Parabolic Problems , 1998 .

[8]  Chuanxi Zhu,et al.  Some results on standing wave solutions for a class of quasilinear Schrödinger equations , 2019, Journal of Mathematical Physics.

[9]  Scipio Cuccagna,et al.  On instability of excited states of the nonlinear Schr , 2008, 0801.4237.

[10]  F. Bass.  Nonlinear electromagnetic-spin waves , 1990 .

[11]  M. Willem.  Minimax Theorems , 1997 .

[12]  Jean Van Schaftingen,et al.  Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics , 2012, 1205.6286.

[13]  Susumu Kurihara,et al.  Large-Amplitude Quasi-Solitons in Superfluid Films , 1981 .

[14]  Xian Wu,et al.  Multiple solutions for quasilinear Schrödinger equations with a parameter , 2014 .

[15]  Olímpio H. Miyagaki,et al.  Soliton solutions for quasilinear Schrödinger equations with critical growth , 2010 .

[16]  Mathieu Colin,et al.  Solutions for a quasilinear Schr"odinger equation: a dual approach , 2004 .

[17]  M. Goldman,et al.  Strong turbulence of plasma waves , 1984 .

[18]  Xianhua Tang,et al.  Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential ☆ , 2014 .