Semiclassical states for non-cooperative singularly perturbed fractional Schrödinger systems

Abstract

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