Multiple solutions for a quasilinear Schrödinger equation. (English) Zbl 1263.35113

Fang, Xiang-Dong; Szulkin, Andrzej

J. Differ. Equations 254, No. 4, 2015-2032 (2013).

Summary: In this paper we consider the quasilinear Schrödinger equation \[ -\Delta u+ V(x)u-\Delta(u^2)u=g(x,u), \qquad x\in \mathbb R^N, \] where \(g\) and \(V\) are periodic in \(x_1, \dots, x_N\) and \(g\) is odd in \(u\), subcritical and satisfies a monotonicity condition. We employ the approach developed by the second author and T. Weth [J. Funct. Anal. 257, No. 12, 3802–3822 (2009; Zbl 1178.35352); The method of Nehari manifold. Somerville, MA: International Press (2010; Zbl 1218.58010)] and obtain infinitely many geometrically distinct solutions.

MSC:

35J62

Quasilinear elliptic equations

Keywords:

quasilinear Schrödinger equation; multiplicity of solutions; Nehari manifold

Citations:

Zbl 1178.35352; Zbl 1218.58010