Multiplicity of solutions for a singular system involving the fractional \(p\)-\(q\)-Laplacian operator and sign-changing weight functions (Q2107774)

Summary: This paper deals with the following singular system: \[ \begin{cases} (-\Delta)_p^s u + (-\Delta)_q^s u = \lambda f (x) |u|^{r-2} u + \frac{1-\alpha}{2-\alpha -\beta} h(x) |u|^{-\alpha} |v|^{1-\beta} &\text{in } \Omega,\\ (-\Delta)_p^s v + (-\Delta)_q^s v = \mu g(x) |v|^{r-2} v +\frac{1-\beta}{2-\alpha -\beta} h(x) |u|^{1-\alpha} |v|^{-\beta} &\text{in } \Omega,\\ u = v = 0 &\text{in } \mathbb{R}^N \setminus \Omega, \end{cases} \] where \(\Omega \subset \mathbb{R}^N\) is a bounded smooth domain, \(\lambda\), \(\mu\) are positive parameters, \(s \in (0,1)\), \(1<p<N/s\), \(0<\alpha\), \(\beta <1\), \(2-\alpha -\beta <q<p<r<p_s^* = N p / (N-sp)\), and \((-\Delta)_\sigma^s u\) denotes the fractional \(\sigma\)-Laplacian, \(\sigma = p\), \(q\). Under appropriate conditions on the weight functions \(f\), \(g\), \(h\) which may change sign in \(\Omega\), we establish the existence of multiple solutions by using the Nehari manifold method. Our paper is one of the first attempts to study the existence of solutions for fractional singular systems involving sign-changing weight functions.