Symmetric and Nonsymmetric Soliton Solutions for a Class of Quasilinear Schrödinger Equations

In this paper we use the variational methods, more precisely, the Mountain-Pass Theorem and Principle of Symmetric Criticality to establish multiplicity of solutions for the following class of quasilinearelliptic problems:\[\begin{array}{lll}-\Delta u +V(z)u-\Delta (u^2)u = h(u), & \;\text{in}\; &\mathbb{R}^N\ \ (N\geq 4).\end{array}\]We assume that the potential $V:\mathbb{R}^N\rightarrow \mathbb{R}$ is positive and bounded away from zero and satisfies periodic and symmetric conditions, and the nonlinear term$h:\mathbb{R}\rightarrow \mathbb{R}$ has subcritical growth and satisfies a condition of the type Ambrosetti-Rabinowitz.