Semiclassical states for non-cooperative singularly perturbed fractional Schrödinger systems

Abstract We study the following non-cooperative type singularly perturbed systems involving fractional Laplacian operator: $$ \textstyle\begin{cases} \varepsilon ^{2s}(-\Delta )^{s} u+a(x)u=g(v), & \text{in } \mathbb{R}^{N}, \\ v+a(x)v=f(u), \end{cases} { ε 2 s ( − Δ ) u + a x = g v , in R N f where $s\in (0,1)$ ∈ 0 1 , $N>2s$ > and $(-\Delta )^{s}$ is s -Laplacian, $\varepsilon >0$ a small parameter. f g are power-type nonlinearities having superlinear subcritical growth at infinity. The corresponding energy functional strongly indefinite, which different from one of single equation case cooperative type. By considering some truncated problems establishing auxiliary results, semiclassical solutions original system obtained using “indefinite theorem”. concentration phenomenon also studied. It shown that can concentrate around global minima potential.