The Nehari Manifold for a Fractional p-Kirchhoff System Involving Sign-Changing Weight Function and Concave-Convex Nonlinearities

摘要

In this paper, we are concernedwith the following fractional p-Kirchhoff system with sign-changing nonlinearities:M(∫)_(R~(2n))(|u(x)? u(y)|~p/|x?y|~(n+ps))dxdy)(?Δ)_p~s u = λa(x)|u|~(q?2)u+(α/(α+β))f(x)|u|~(α?2)u|v|~β, in Ω,M(∫_(R~(2n)) (|v(x)?v(y)|~p/|x?y|~(n+ps))dxdy)(?Δ)_p~s v =μb(x)|V|~(q?2)v + (β/(α + β))f(x)|u|~α|v|~(β?2)v, in Ω, and u = v = 0, in R~n Ω, where Ω is a smooth bounded domain in R~n, n > ps, s ∈ (0,1), λ, μ are two real parameters, 1 < q < p < p(h + 1) < α + β < p_s~? = np/(n ? ps), M is a continuous function, given by M(t) = k + lt~h, k > 0, l > 0, h ≥ 1,a(x),b(x) ∈ L~((α+β)/(α+β?q))(Ω) are sign changing and either a~± = max{±a, 0} ≠ 0 or b~± = max{±b, 0} ≠ 0, f ∈ L(Ω-bar) with ‖f‖_∞ = 1, and f ≥ 0. Using Nehari manifold method, we prove that the system has at least two solutions with respect to the pair of parameters (λ, μ).