On the existence of nontrivial solutions for quasilinear Schrödinger systems

Abstract

Abstract In this paper, by using a change of variable and the mountain-pass theorem, we show that the following quasilinear Schrödinger systems { − △ u + V 1 ( x ) u + κ 2 [ △ | u | 2 ] u = λ f ( x , u , v ) , x ∈ R N , − △ v + V 2 ( x ) v + κ 2 [ △ | v | 2 ] v = λ h ( x , u , v ) , x ∈ R N $$ \textstyle\begin{cases} -\triangle u+V_{1}(x)u+\frac{\kappa}{2}[\triangle \vert u \vert ^{2}]u=\lambda f(x,u, v), & x\in \mathbb{R}^{N}, \\ -\triangle v+V_{2}(x)v+\frac{\kappa}{2}[\triangle \vert v \vert ^{2}]v=\lambda h(x,u, v), & x\in \mathbb{R}^{N} \end{cases} $$ have a nontrivial solution ( u , v ) $(u, v)$ for all λ > λ 1 ( κ ) $\lambda >\lambda _{1}(\kappa )$ , where N ≥ 3 , V 1 ( x ) , V 2 ( x ) $N\geq 3, V_{1}(x), V_{2}(x)$ are positive continuous functions, κ, λ are positive parameters, and nonlinear terms f , h ∈ C ( R N × R 2 , R ) $f, h\in C(\mathbb{R}^{N}\times \mathbb{R}^{2}, \mathbb{R})$ . Our main contribution is that we can deal with the case when κ > 0 $\kappa >0$ is large for the above systems.

Keywords

• Quasilinear Schrödinger systems
• Mountain-pass theorem
• Morse iteration
• Nontrivial solution