On the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity

Abstract

In this paper, we consider the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity ε2sM([u]s,Aε2)(−Δ)Aεsu+V(x)u=|u|2s∗−2u+h(x,|u|2)u,x∈RN,u(x)→0,as|x|→∞,where (−Δ)Aεs is the fractional magnetic operator with 0<s<1, 2s∗=2N∕(N−2s), M:R0+→R+ is a continuous nondecreasing function, V:RN→R0+ and A:RN→RN are the electric and magnetic potentials, respectively. By using the fractional version of the concentration compactness principle and variational methods, we show that the above problem: (i) has at least one solution provided that ε<E; and (ii) for any m∗∈N, has m∗ pairs of solutions if ε<Em∗, where E and Em∗ are sufficiently small positive numbers. Moreover, these solutions uε→0 as ε→0.