Existence of Ground States of Fractional Schrödinger Equations

We consider ground states of the nonlinear fractional Schrödinger equation with potentials ( - Δ ) s u + V ( x ) u = f ( x , u ) , s ∈ ( 0 , 1 ) , (-\Delta)^{s}u+V(x)u=f(x,u),\quad s\in(0,1), on the whole space ℝ N {\mathbb{R}^{N}} , where V is a periodic non-negative nontrivial function on ℝ N {\mathbb{R}^{N}} and the nonlinear term f has some proper growth on u . Under uniform bounded assumptions about V , we can show the existence of a ground state. We extend the result of Li, Wang, and Zeng to the fractional case.

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