Fractional p-Kirchhoff system with sign changing nonlinearities

Abstract

In this paper, we show the existence and multiplicity of nontrivial, non-negative solutions of the following fractional p-Kirchhoff system

$$\begin{aligned}&M\left( \displaystyle \int _{\mathbb {R}^{2n}}\frac{|u(x)-u(y)|^p}{\left| x-y\right| ^{n+ps}}dx\,dy\right) (-\Delta )^{s}_p u(x) \\&\quad = \lambda f(x)|u|^{q-2}u+ \frac{2\alpha }{\alpha +\beta }\left| u\right| ^{\alpha -2}u|v|^\beta \quad \text {in } \Omega ,\\&M\left( \displaystyle \int _{\mathbb {R}^{2n}}\frac{|v(x)-v(y)|^p}{\left| x-y\right| ^{n+ps}}dx\,dy\right) (-\Delta )^{s}_p v(x) \\&\quad =\mu g(x)|v|^{q-2}v+ \frac{2\beta }{\alpha +\beta }\left| u\right| ^{\alpha }|v|^{\beta -2}v\quad \text {in } \Omega ,\\&u = v = 0 \quad \text {in }\; \mathbb {R}^{n}{\setminus }\Omega , \end{aligned}$$

where $(-\Delta )^{s}_p$ is the fractional p-Laplace operator, $\Omega$ is a bounded domain in $\mathbb {R}^n$ with smooth boundary, M is continuous function, $\lambda , \mu$ are real parameters, $f,g \in L^{\gamma }(\Omega )$ with $\gamma =\frac{\alpha +\beta }{\alpha +\beta -q}$ are sign changing, $ps<n<2ps$ and $1<q<p$, $2p<r\le p_s^*=\frac{np}{n-ps}$.