Existence and properties of soliton solution for the quasilinear Schrödinger system

Abstract

In this article, we consider the following quasilinear Schrödinger system: − ε Δ u + u + k 2 ε [ Δ ∣ u ∣ 2 ] u = 2 α α + β ∣ u ∣ α − 2 u ∣ v ∣ β , x ∈ R N , − ε Δ v + v + k 2 ε [ Δ ∣ v ∣ 2 ] v = 2 β α + β ∣ u ∣ α ∣ v ∣ β − 2 v , x ∈ R N , \left\{\begin{array}{ll}-\varepsilon \Delta u+u+\frac{k}{2}\varepsilon \left[\Delta \hspace{-0.25em}{| u| }^{2}]u=\frac{2\alpha }{\alpha +\beta }{| u| }^{\alpha -2}u{| v| }^{\beta },& x\in {{\mathbb{R}}}^{N},\\ -\varepsilon \Delta v+v+\frac{k}{2}\varepsilon \left[\Delta \hspace{-0.25em}{| v| }^{2}]v=\frac{2\beta }{\alpha +\beta }{| u| }^{\alpha }{| v| }^{\beta -2}v,& x\in {{\mathbb{R}}}^{N},\end{array}\right. where ε > 0 , k < 0 \varepsilon \gt 0,k\lt 0 are real constants, N ≥ 3 N\ge 3 , α , β \alpha ,\beta are integers multiple of constant 2. By using the Mountain Pass Theorem in a suitable Orlicz space proposed by Abbas Moameni [Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in R N {{\mathbb{R}}}^{N} , J. Differential Equations 229 (2006), 570–587], we proved the existence of soliton solution ( u ε , v ε ) \left({u}_{\varepsilon },{v}_{\varepsilon }) for the above system, and ( u ε ( x ) , v ε ( x ) ) → ( 0 , 0 ) ({u}_{\varepsilon }\left(x),{v}_{\varepsilon }\left(x))\to \left(0,0) as ∣ ε ∣ → 0 | \varepsilon | \to 0 .