Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 24, pp. 1-21.

Title: Existence of positive solutions for systems of quasilinear Schrodinger equations

Authors: Ayesha Baig (Central South Univ., Changsha, China)
         Zhouxin Li  (Central South Univ., Changsha, China)
  
Abstract:
In this article, we study the existence of standing wave solutions 
for the  quasilinear Schrodinger system
$$\displaylines{
- \varepsilon^2 \Delta u + W(x) u - \kappa \varepsilon^2 \Delta (u^2) u 
= Q_u (u,v) \quad \text{in } \mathbb{R}^N, \cr
- \varepsilon^2 \Delta v + V(x) v - \kappa \varepsilon^2 \Delta (v^2) v 
= Q_v (u,v) \quad \text{in } R^N, \cr
u, v > 0 \quad \text{in } R^N, \quad u,v \in H^1 (R^N).
}$$
where $N \geq 3$, $\kappa > 0$, $\varepsilon > 0$, 
$W,V:\mathbb{R}^N \to \mathbb{R}$ are continuous functions that fall
into two classes of potentials. To overcome the lack of differentiability, 
we use the dual approach developed by Colin–Jeanjean. 
The existence of solutions is obtained using Del Pino–Felmer’s penalization 
technique with an adaptation of Alves’ arguments [1].

Submitted April 17, 2024. Published March 03, 2025.
Math Subject Classifications: 35Q55, 58E05, 58E30.
Key Words: Quasilinear Schrodinger system; Cerami sequence; dual approach; positive solution.