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

Quanqing Li ^[1] ; Jianjun Nie ^[3] ; Wenbo Wang ^[2]
1. [1] Honghe University
  
  Honghe University
  
  China
2. [2] Yunnan University
  
  Yunnan University
  
  China
3. [3] North China Electric Power University
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Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 2, 2024
Idioma: inglés
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Resumen
- 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.