Abstract
In this work we consider the fractional Kirchhoff equations with singular nonlinearity, $$\displaylines{ M\Big( \int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dx dy\Big) (-\Delta)^s_p u = \lambda a(x)|u|^{q-2}u +\frac{1-\alpha}{2-\alpha-\beta} c(x)|u|^{-\alpha}|v|^{1-\beta}, \quad \text{in }\Omega,\cr M\Big( \int_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^p}{|x-y|^{N+sp}}dx dy\Big) (-\Delta)^s_p v = \mu b(x)|v|^{q-2}v +\frac{1-\beta}{2-\alpha-\beta} c(x)|u|^{1-\alpha}|v|^{-\beta}, \quad \text{in }\Omega, \cr u=v = 0 ,\quad\hbox{in }\mathbb{R}^N\setminus\Omega, }$$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary, \(N> ps\), \(s \in (0,1)\), \(0<\alpha<1\), \(0<\beta<1\), \(2-\alpha-\beta<p\leq p\theta<q<p^*_s\), \(p^*_s=\frac{Np}{N-sp}\) is the fractional Sobolev exponent, \(\lambda, \mu\) are two parameters, \(a, b, c \in C(\overline{\Omega})\) are non-negative weight functions, \(M(t)=k+lt^{\theta-1}\) with \(k>0,l,\theta\geq 1\), and \((-\Delta)^s_p\) is the fractional p-laplacian operator. We prove the existence of multiple non-negative solutions by studying the nature of the Nehari manifold with respect to the parameters \(\lambda\) and \(\mu\).