Uniqueness of weak solution for nonlocal \((p,q)\)-Kirchhoff system

The paper aims to investigate the existence and uniqueness of weak solution, using the Browder Theorem method, for the nonlocal \((p,q)\)-Kirchhoff system: \[\begin{cases} -K_{1}\big(\int_{\Omega}|\nabla \phi|^{p}\big)\Delta_{p}\phi+\lambda a(x)|\phi|^{p-2}\phi=f_1(x,\phi,\psi), & x\in \Omega \\ -K_{2}\big(\int_{\Omega}|\nabla \psi|^{q}\big)\Delta_{q}\psi+\lambda b(x)|\psi|^{q-2}v=f_2(x,\phi,\psi), & x\in \Omega \\ \phi=\psi=0, & x \in \partial\Omega \end{cases}\] where \(\Omega\) is a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary \(\partial\Omega\), with \(K_{1},K_{2}\) be continuous functions and \(f_1,f_2\) be Carathéodory functions.