Multiplicity of solutions for a class of fractional p-Kirchhoff system with sign-changing weight functions

Abstract

Abstract In this paper, we investigate the fractional p-Kirchhoff -type system: {M(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)(−Δ)psu=μg(x)|u|β−2u+aa+bh(x)|u|a−2u|v|b,in Ω,M(∫R2N|v(x)−v(y)|p|x−y|N+psdxdy)(−Δ)psv=σf(x)|v|β−2v+ba+bh(x)|v|b−2v|u|a,in Ω,u=v=0,in RN∖Ω, $$\begin{aligned} \textstyle\begin{cases} M (\int_{{ \mathbb {R} }^{2N}}\frac{\vert u(x)-u(y) \vert ^{p}}{\vert x-y \vert ^{N+ps}}\,dx\,dy )(- \Delta )^{s}_{p}u=\mu g(x)\vert u \vert ^{\beta -2}u+\frac{a}{a+b}h(x)\vert u \vert ^{a-2}u\vert v \vert ^{b},&\mbox{in } \Omega , \\ M (\int_{{ \mathbb {R} }^{2N}}\frac{\vert v(x)-v(y) \vert ^{p}}{\vert x-y \vert ^{N+ps}}\,dx\,dy )(- \Delta )^{s}_{p}v=\sigma f(x)\vert v \vert ^{\beta -2}v+\frac{b}{a+b}h(x)\vert v \vert ^{b-2}v\vert u \vert ^{a},&\mbox{in } \Omega , \\ u=v=0,&\mbox{in } { \mathbb {R} }^{N}\setminus \Omega , \end{cases}\displaystyle \end{aligned}$$ where Ω⊂RN $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain, (−Δ)ps $(-\Delta )^{s}_{p}$ is the fractional p-Laplacian operator with 01 $a>1$, b>1 $b>1$ satisfy 20 $k>0$, λ, τ≥0 $\tau \geq 0$, τ=0 $\tau =0$ if and only if λ=0 $\lambda =0$. The weight functions g, f, h change sign in Ω and satisfy suitable conditions. By using the Nehari manifold method, it is proved that the system has at least two solutions provided that 2λ1 $\lambda >\lambda_{1}$ under the assumptions μ=σ=0 $\mu =\sigma =0$ and p<a+b<min{p(τ+1),ps∗} $p< a+b<\min \{p(\tau +1),p_{s}^{*}\}$.

Keywords

• Fractional p-Kirchhoff system
• Multiplicity
• Sign-changing weight functions
• Nehari manifold
• Mountain pass theorem