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

In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical growth \(\begin{aligned} (-\Delta )_A^su+V(x)u=\lambda |u|^{p-2}u+ f(x,|u|^2)u, \ x \in \mathbb {R}^N, \end{aligned}\) $$\begin{array}{c}\hfill {(-\mathrm{\Delta })}_A^{s}u+V(x)u={\lambda |u|}^{p-2}u+{f(x,|u|}^2)u,x\in {\mathbb{R}}^N,\end{array}$$ where \((-\Delta )_A^s\) ${(-\mathrm{\Delta })}_A^s$ is the fractional magnetic operator with \(0<s<1\) $0<s<1$ , \(N>2s\) $N>2s$ , \(2_s^*=\frac{2N}{N-2s}\) $2_s^{\ast }=\frac{2N}{N-2s}$ , \(\lambda >0\) $\lambda >0$ , \(V \in C(\mathbb {R}^N,\mathbb {R})\) $V\in C({\mathbb{R}}^N,\mathbb{R})$ and \(A \in C(\mathbb {R}^N, \mathbb {R}^N)\) $A\in C({\mathbb{R}}^N,{\mathbb{R}}^N)$ 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 \(2< q<2_s^*\) $2<q<2_s^{\ast }$ such that \(|f(x,t)|\le C(1+|t|^{\frac{q-2}{2}})\) $|f(x,t)|\le C(1+|t{|}^{\frac{q-2}{2}})$ for all (x, t), for \( 2_s^*\le p<22^{*}_{s}-q\) $2_s^{\ast }\le p<{22}_s^{\ast }-q$ . For any \(D>0\) $D>0$ fixed, if \(\lambda \in (0,D]\) $\lambda \in (0,D]$ we prove that the equation has a nontrivial solution by the truncation method. Our method can provide a prior \(L^{\infty }\) $L^{\infty }$ -estimate.