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

Abstract

In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical growth

$$\begin{aligned} (-\Delta )_A^su+V(x)u=\lambda |u|^{p-2}u+ f(x,|u|^2)u, \ x \in \mathbb {R}^N, \end{aligned}$$

where $(-\Delta )_A^s$ is the fractional magnetic operator with $0<s<1$, $N>2s$, $2_s^*=\frac{2N}{N-2s}$, $\lambda >0$, $V \in C(\mathbb {R}^N,\mathbb {R})$ and $A \in C(\mathbb {R}^N, \mathbb {R}^N)$ 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 $2< q<2_s^*$ such that $|f(x,t)|\le C(1+|t|^{\frac{q-2}{2}})$ for all (x, t), for $2_s^*\le p<22^{*}_{s}-q$. For any $D>0$ fixed, if $\lambda \in (0,D]$ we prove that the equation has a nontrivial solution by the truncation method. Our method can provide a prior $L^{\infty }$-estimate.

Appendix

In the present paper, we have constructed a $C^{1}$ function $\gamma (\cdot )$ which remedies the weakness in our paper [21, Lemma 2.2]. It is worth mentioning that in the present paper, we have resurfaced our results involving critical or supercritical results. And we caution readers that the truncation function $\phi _{M}$ depends on M and some previous results (including [21, 29] and so on) need to be stated by another way. The authors thank the anonymous researchers for their useful discussions.