Multiple solutions to critical magnetic Schrödinger equations

In this paper, we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation (0.1) \((-{\rm i}\nabla-A(x))^2u+\lambda V(x)u=\mu \vert u\vert ^{p-2}u+\Big(\int_{\mathbb{R^N}}\frac{\vert u(y)\vert ^{2^*_\alpha}}{\vert x-y\vert ^\alpha}{\rm d}y\Big)\vert u\vert ^{2^*_\alpha-2}u\quad {\rm in}\ \mathbb{R}^N,\) $$(-\mathrm{i}\mathrm{\nabla }-A(x){)}^{2}u+\lambda V(x)u=\mu \mid u{\mid }^{p-2}u+({\int }_{{\mathbb{R}}^{\mathbb{N}}}\frac{\mid u(y){\mid }^{2_{\alpha }^{\ast }}}{\mid x-y{\mid }^{\alpha }}\mathrm{d}y)\mid u{\mid }^{2_{\alpha }^{\ast }-2}u\mathrm{i}n{\mathbb{R}}^N,$$

where \(N\geq4, 2\leq p<2^*, 2^*_\alpha=\frac{2N-\alpha}{N-2}\) $$N\ge 4,2\le p<2^{\ast },2_{\alpha }^{\ast }=\frac{2N-\alpha }{N-2}$$ with 0 < α < 4, λ > 0, μ ∈ ℝ, A(x) = (A₁ (x), A₂ (x), ⋯, A_N (x)) is a real local Hölder continuous vector function, i is the imaginary unit, and V(x) is a real valued potential function on ℝ^N. Supposing that Ω = int V⁻¹ (0) ⊂ ℝ^N is bounded, we show that problem (0.1) possesses at least cat_Ω(Ω) nontrivial solutions if λ is large.