Ground states and concentration phenomena for the fractional Schrödinger equation

We consider here solutions of the nonlinear fractional Schrödinger equation $ε^{2s}(-Δ)^s u+V(x)u=u^p.$ We show that concentration points must be critical points for $V$. We also prove that, if the potential $V$ is coercive and has a unique global minimum, then ground states concentrate suitably at such minimal point as $ε$ tends to zero. In addition, if the potential $V$ is radial, then the minimizer is unique provided $ε$ is small.

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