Existence and uniqueness of solutions for the $p$-Lamé Dirichlet problem by topological degree

We consider a mathematical model named the generalized Lamé system ($p$-Lamé), which describes the displacement $u$ from the natural state of a nonhomogeneous elastic solid subjected to a volume density of forces $f$ that depends on the displacement $u$ in a domain $\Omega $ of $\mathbb {R}^{N}$. Using the topological degree theory for a class of demicontinuous operators of generalized $(S_{+})$ type, we prove the existence and uniqueness of the weak solution.