Existence and multiplicity of solutions to a $p-q$ Laplacian system with a concave and singular nonlinearities

Abstract

In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -\Delta_{p}u-\Delta_q u &= \lambda f(x)|u|^{r-2}u+\nu\frac{1-\alpha}{2-\alpha-\beta}h(x) |u|^{-\alpha}|v|^{1-\beta}\,\,\mbox{in}\,\,\Omega,\\ -\Delta_{p}v-\Delta_q v &= \mu g(x)|v|^{r-2}v+\nu\frac{1-\beta}{2-\alpha-\beta}h(x) |u|^{1-\alpha}|v|^{-\beta}\,\,\mbox{in}\,\,\Omega,\\ u,v&>0\,\,\mbox{in}\,\,\Omega,\\ u= v &= 0\,\, \mbox{on}\,\, \partial\Omega \end{split} \end{align*} where (C):~$0<\alpha<1,\;0<\beta<1,$ $2-\alpha-\beta<q<\frac{N(p-1)}{N-p}<p<r<p^*$, with $p^*=\frac{Np}{N-p}$. We will guarantee the existence of a solution in the Nehari manifold. Further by using the Lusternik-Schnirelman category we will prove the existence of at least $\text{cat}(\Omega)+1$ number of solutions.