Positive solutions for a class of quasilinear Schrödinger equations with superlinear condition

Abstract In this paper, we study the following quasilinear Schrodinger equation − Δ u + V ( x ) u + κ 2 Δ ( u 2 ) u = f ( u ) , x ∈ R N , where N ≥ 3 , κ > 0 and V : R N → R satisfies suitable assumptions. By using a change of variable, we obtain the existence of positive solutions, which use the method developed by Jeanjean (1999).

[1]  Edcarlos D. Silva,et al.  On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms , 2017 .

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

[3]  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.

[4]  Jianhua Chen,et al.  Existence of ground state solutions for quasilinear Schrödinger equations with super-quadratic condition , 2018, Appl. Math. Lett..

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

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

[7]  M. Souto,et al.  Equation with positive coefficient in the quasilinear term and vanishing potential , 2015 .

[8]  C. O. Alves,et al.  Soliton solutions for a class of quasilinear Schrödinger equations with a parameter , 2013, 1309.4606.

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

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

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

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

[13]  Minbo Yang,et al.  Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity , 2019, Communications in Contemporary Mathematics.

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

[15]  Yaotian Shen,et al.  Soliton solutions for generalized quasilinear Schrödinger equations , 2013 .

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