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

Abstract

In this paper, we are concernedwith the following fractional p-Kirchhoff systemwith sign-changing nonlinearities:M(∫ R2n (|u(x)− u(y)|p/|x−y|n+ps)dxdy)(−Δ)spu = λa(x)|u|q−2u+(α/(α+β))f(x)|u|α−2u|V|β, in Ω,M(∫R2n (|V(x)−V(y)|p/|x−y|n+ps)dxdy)(−Δ)spV = μb(x)|V|q−2V + (β/(α + β))f(x)|u|α|V|β−2V, in Ω, and u = V = 0, in R \\ Ω, where Ω is a smooth bounded domain in R, 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 + lth, 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) 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).