Resumen de Nontrivial Solutions for Fractional Schrödinger Equations with Electromagnetic Fields and Critical or Supercritical Growth

Quanqing Li, Jianjun Nie, Wenbo Wang

• In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical growth (−)s Au + V(x)u = λ|u| p−2u + f (x, |u| 2)u, x ∈ RN , where (−)s A is the fractional magnetic operator with 0 < s < 1, N > 2s, 2∗ s = 2N N−2s , λ > 0, V ∈ C(RN , R) and A ∈ C(RN , RN ) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, and f is a continuous function and there exists 2 < q < 2∗ s such that | f (x, t)| ≤ C(1+|t| q−2 2 ) for all (x, t), for 2∗ s ≤ p < 22∗ s − q. For any D > 0 fixed, if λ ∈ (0, D] we prove that the equation has a nontrivial solution by the truncation method. Our method can provide a prior L∞-estimate.