Existence of nontrivial solutions for fractional Schrödinger equations with electromagnetic fields and critical or supercritical nonlinearity

• Main Authors: Quanqing Li, Kaimin Teng, Wenbo Wang, Jian Zhang
• Format: Article
• Language: English
• Published: SpringerOpen 2020-06-01
• Series: Boundary Value Problems
• Subjects: Fractional Schrödinger equation; Fractional magnetic operator; Critical or supercritical nonlinearity
• Online Access: http://link.springer.com/article/10.1186/s13661-020-01409-1

Abstract In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity: ( − Δ ) A s u + V ( x ) u = f ( x , | u | 2 ) u + λ | u | p − 2 u , x ∈ R N , $$ (-\Delta )_{A}^{s}u+V(x)u=f\bigl(x, \vert u \vert ^{2}\bigr)u+\lambda \vert u \vert ^{p-2}u,\quad x \in \mathbb{R}^{N}, $$ where ( − Δ ) A s $(-\Delta )_{A}^{s}$ is the fractional magnetic operator with 0 < s < 1 $0< s<1$ , N > 2 s $N>2s$ , λ > 0 $\lambda >0$ , 2 s ∗ = 2 N N − 2 s $2_{s}^{*}=\frac{2N}{N-2s}$ , p ≥ 2 s ∗ $p\geq 2_{s}^{*}$ , f is a subcritical nonlinearity, and V ∈ C ( R N , R ) $V \in C(\mathbb{R}^{N},\mathbb{R})$ and A ∈ C ( R N , R N ) $A \in C(\mathbb{R}^{N}, \mathbb{R}^{N})$ are the electric and magnetic potentials, respectively. Under some suitable conditions, by variational methods we prove that the equation has a nontrivial solution for small λ > 0 $\lambda >0$ . Our main contribution is related to the fact that we are able to deal with the case p > 2 s ∗ $p>2_{s}^{*}$ .