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

Jiabin Zuo,Tianqing An,Libo Yang,Xiangsheng Ren

doi:10.1155/2019/7624373

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).

Highlights

In this article, we study the following fractional p-Kirchhoff problem (P) involving concave-convex nonlinearities and sign-changing weight functions: M (∫ R2n󵄨󵄨󵄨󵄨u (x) − u (y)󵄨󵄨󵄨󵄨p 󵄨󵄨󵄨󵄨x − y󵄨󵄨󵄨󵄨n+ps dxdy) (−Δ)sp u = λa (x) |u|q−2 u + α α + βf (x)
Using Nehari manifold method, we prove that the system has at least two solutions with respect to the pair of parameters (λ, μ)
We study the following fractional p-Kirchhoff problem (P) involving concave-convex nonlinearities and sign-changing weight functions: M

Summary

Introduction

We study the following fractional p-Kirchhoff problem (P) involving concave-convex nonlinearities and sign-changing weight functions:. There are many excellent and interesting results about the existence and multiplicity of solutions for nonlocal fractional problems. For a single equation with sign-changing weights functions, in [9, 10], the authors studied the existence and multiplicity of nonnegative solutions in subcritical and critical cases respectively. In the special case of p = 2, s = 1, and M = 1, Tsung-Fang Wu [11] proved that system has least two nontrivial nonnegative solutions by using the Nehari manifold. For a more general case M = k + ltτ, Yang and An [13] show the system has at least two solutions with the help of Nehari manifold, but without considering signchanging weights functions.

The Variational Setting and Preliminaries

Proof of the Main Result