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

Abstract

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 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 ∗ $p>2_{s}^{*}$ .

Keywords

• Fractional Schrödinger equation
• Fractional magnetic operator
• Critical or supercritical nonlinearity