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

Abstract

In this paper, we are concerned with the following fractional p-Kirchhoff system with sign-changing nonlinearities: M(∫R2nux-uyp/x-yn+psdxdy)-Δpsu=λa(x)uq-2u+α/(α+β)f(x)uα-2uvβ,  in  Ω, M(∫R2n|v(x)-v(y)|p/|x-y|n+psdxdy)-Δpsv=μb(x)vq-2v+(β/α+β)f(x)uαvβ-2v,  in  Ω, and u=v=0,  in  Rn∖Ω, where Ω is a smooth bounded domain in Rn, n>ps, s∈(0,1), λ, μ are two real parameters, 1<q<p<p(h+1)<α+β<ps⁎=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).