We study the following fractional Choquard equation ε 2 s ( − ∆ ) s u + V ( x ) u = ε − µ ( K µ ∗ F ( u ) ) F ′ ( u ) , x ∈ R N , where ε>0 is a small parameter, s∈(0 ,1), N⩾3, µ∈(0 ,N), F ∈ C 1 ( R , R ) , K µ is the Riesz potential. By applying a new variational approach, under some appropriate conditions on V( x), we obtain there exist at least cupl ( V ) + 1 solutions to the above equation when ϵ→0. In addition, we have demonstrated that the concentration behavior of positive solutions occurs around V as ϵ→0, where V is the set where the potential attains its minimum values.

The aim of this paper is to study the existence and multiplicity of nonnegative solutions for the following critical Kirchhoff equation involving the fractional p-Laplace operator ( − Δ)ps. More precisely, we consider $$ M\left(^{2N}}}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s} u=\lambda f(x)|u|^{q-2}u+K(x)|u|^{p_{s}^{*}-2}u,\quad &{\rm in}\ \Omega,\\ u=0, \quad\quad &{\rm in}\ ^{N}\setminus \Omega, \\ $$ where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary ∂Ω, M(t)=a + btm − 1 with m > 1, a > 0, b > 0, dimension N > sp, $ p_{s}^{*}={N-ps}$ is the fractional critical Sobolev exponent, and the parameters λ > 0, 0 < s < 1 < q < p < ∞. Applying Nehari manifold, fibering maps and Krasnoselskii genus theory, we investigate the existence and multiplicity of nonnegative solutions.