Multiple solutions for a fractional p-Kirchhoff problem with Hardy nonlinearity. (November 2019) - Chen, Wenjing; Gui, Yuyan
Abstract: This paper is devoted to study the existence of multiple solutions for the following fractional p -Kirchhoff problem (0.1) M ∫ R 2 n | u ( x ) − u ( y ) | p | x − y | n + p s d x d y ( − △ ) p s u = λ | u | q − 2 u + | u | r − 2 u | x | α, in Ω, u = 0, in R n ∖ Ω, where ( − △ ) p s denotes the fractional p -Laplace operator, Ω is a smooth bounded set in R n containing 0 with Lipschitz boundary, M ( t ) = a + b t θ − 1 with a ≥ 0, b > 0, θ > 1 . λ > 0, 1 < q < p < θ p ≤ r ≤ p α ∗, p α ∗ = ( n − α ) p n − p s is the fractional critical Hardy–Sobolev exponent for 0 ≤ α < p s < n . By using fibering maps and Nehari manifold, we obtain that the existence of multiple solutions to problem (0.1) for both Hardy–Sobolev subcritical and critical cases. In particular, the concentration compactness principle will be used to overcome the lack of compactness for the critical case.