Existence and properties of soliton solution for the quasilinear Schrödinger system

Abstract

In this article, we consider the following quasilinear Schrödinger system: −εΔu+u+k2ε[Δ∣u∣2]u=2αα+β∣u∣α−2u∣v∣β,x∈RN,−εΔv+v+k2ε[Δ∣v∣2]v=2βα+β∣u∣α∣v∣β−2v,x∈RN,\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≥3N\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 RN{{\mathbb{R}}}^{N} , J. Differential Equations 229 (2006), 570–587], we proved the existence of soliton solution (uε,vε)\left({u}_{\varepsilon },{v}_{\varepsilon }) for the above system, and (uε(x),vε(x))→(0,0)({u}_{\varepsilon }\left(x),{v}_{\varepsilon }\left(x))\to \left(0,0) as ∣ε∣→0| \varepsilon | \to 0.

Keywords

• orlicz space
• quasilinear schrödinger system
• mountain pass theorem
• 35j10
• 35j50
• 35c08