Ground states for fractional Choquard equations with magnetic fields and critical exponents

Abstract

Abstract In this paper, we investigate the ground states for the following fractional Choquard equation with magnetic fields and critical exponents: ( - Δ ) A s ⁢ u + V ⁢ ( x ) ⁢ u = λ ⁢ f ⁢ ( x , u ) + [ | x | - α ∗ | u | 2 α , s * ] ⁢ | u | 2 α , s * - 2 ⁢ u in ⁢ ℝ N , (-\Delta)_{A}^{s}u+V(x)u=\lambda f(x,u)+[\lvert x\rvert^{-\alpha}\ast\lvert u% \rvert^{2^{*}_{\alpha,s}}]\lvert u\rvert^{2^{*}_{\alpha,s}-2}u\quad\text{in }% \mathbb{R}^{N}, where λ > 0 {\lambda>0} , α ∈ ( 0 , 2 ⁢ s ) {\alpha\in(0,2s)} , N > 2 ⁢ s {N>2s} , u : ℝ N → ℂ {u:\mathbb{R}^{N}\rightarrow\mathbb{C}} is a complex-valued function, 2 α , s * = ( 2 ⁢ N - α ) / ( N - 2 ⁢ s ) {2^{*}_{\alpha,s}=(2N-\alpha)/(N-2s)} is the fractional Hardy–Littlewood–Sobolev critical exponent, V ∈ ( ℝ N , ℝ ) {V\in(\mathbb{R}^{N},\mathbb{R})} is an electric potential, V and f are asymptotically periodic in x, A ∈ ( ℝ N , ℝ N ) {A\in(\mathbb{R}^{N},\mathbb{R}^{N})} is a magnetic potential, and ( - Δ ) A s {(-\Delta)^{s}_{A}} is a fractional magnetic Laplacian operator with s ∈ ( 0 , 1 ) {s\in(0,1)} . We prove that the equation has a ground state solution for large λ by using the Nehari method and the concentration-compactness principle.