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

• Main Authors: Yunfeng Wei, Caisheng Chen, Hongwei Yang, Hongxue Song
• Format: Article
• Language: English
• Published: SpringerOpen 2018-05-01
• Series: Boundary Value Problems
• Subjects: Fractional p-Kirchhoff system; Multiplicity; Sign-changing weight functions; Nehari manifold; Mountain pass theorem
• Online Access: http://link.springer.com/article/10.1186/s13661-018-0998-7

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 0<s<1<p $0< s<1<p$ and ps<N $ps< N $. a>1 $a>1$, b>1 $b>1$ satisfy 2<a+b<ps∗ $2< a+b< p_{s}^{*}$. 1<β<ps∗ $1<\beta <p_{s}^{*}$, ps∗=NpN−ps $p_{s}^{*}=\frac{Np}{N-ps}$ is the fractional critical exponent. μ, σ are two real parameters. M(t)=k+λtτ $M(t)=k+\lambda t^{\tau }$, k>0 $k>0$, 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 2<a+b<p≤p(τ+1)<β<ps∗ $2< a+b< p\leq p(\tau +1)<\beta <p_{s}^{*}$ and (μ,σ) $(\mu ,\sigma )$ belongs to a certain subset of R2 $\mathbb {R} ^{2}$. Also, by using the mountain pass theorem, we prove that there exist λ1≥λ0 $\lambda _{1}\geq \lambda_{0}$ such that the system admits at least a nontrivial solution for λ∈(0,λ0) $\lambda \in (0,\lambda_{0})$ and no nontrivial solution for λ>λ1 $\lambda >\lambda_{1}$ under the assumptions μ=σ=0 $\mu =\sigma =0$ and p<a+b<min{p(τ+1),ps∗} $p< a+b<\min \{p(\tau +1),p_{s}^{*}\}$.