Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in RN - Pucci, Patrizia; Xiang, Mingqi; Zhang, Binlin (Calculus of variations and partial differential equations, 2015)
In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p -Laplacian equations of Schrödinger–Kirchhoff type M ∫ ∫ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y ( - Δ ) p s u + V ( x ) | u | p - 2 u = f ( x , u ) + g ( x ) in R N , where ( - Δ ) p s is the fractional p -Laplacian operator, with 0 < s < 1 < p < ∞ and p s < N , the nonlinearity f : R N × R → R is a Carathéodory function and satisfies the Ambrosetti–Rabinowitz condition, V : R N → R + is a potential function and g : R N → R is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.