Abstract:

In this paper, we consider the existence of multiple solutions of the following multi-critical nonlocal elliptic equations with magnetic field

where $\Omega$ is bounded domain with smooth boundary in $\mathbb{R}^N$, $N\geq4$, i is imaginary unit, $2^*_s=\frac{N+\alpha_s}{N-2}$ with $N-4$<$\alpha_s$<$N, s=1,2,\cdots,k$ $(k\geq2)$, $\lambda$>0 and $2\leq p$<$2^*=\frac{2N}{N-2}$. Suppose the magnetic vector potential $A(x)= (A_1(x), A_2(x),\cdots, A_N(x))$ is real and local Hölder continuous, we show by the Ljusternik-Schnirelman theory that our problem has at least ${\rm cat}_\Omega(\Omega)$ nontrivial solutions for $\lambda$ small.

Key words: Multi-critical elliptic problem, Magnetic potential, Ljusternik-Schnirelman theory