Existence results for Navier problems with degenerated (p,q)-Laplacian and (p,q)-Biharmonic operators

Abstract

In this article, we prove the existence and uniqueness of solutions for the Navier problem \[ (P)\left\{ \begin{array}{llll} & {\Delta}{\big[}{\omega}(x)(\,{\vert{\Delta}u\vert}^{p-2}{\Delta}u + {\vert{\Delta}u\vert}^{q-2}{\Delta}u){\big]} -{\rm div}{\big[}{\omega}(x)(\,{\vert{\nabla}u\vert}^{p-2}{\nabla}u + {\vert{\nabla}u\vert}^{q-2}{\nabla}u){\big]}\\ & = f(x) - {\rm div}(G(x)),\ \ {\rm in} \ \ {\Omega}, \\ & u(x) = {\Delta}u= 0, \ \ {\rm in} \ \ {\partial\Omega}, \end{array} \right. \] \noindent where Ω is a bounded open set of ${\real}^N$ (N≥2), $\displaystyle {\dfrac{f}{\omega}}\,{\in}\,L^{p\,'}(\Omega , \omega)$ and $\displaystyle{\dfrac{G}{\omega}}\, {\in}\,[L^{q\,'}(\Omega , \omega)]^N$ .

Keywords

• degenerate nonlinear elliptic equations
• weighted sobolev space