Existence result for double phase problem involving the (p(x),q(x))-Laplacian-like operators

The paper study the existence of at least one weak solutions for Dirichlet boundary value problem involving the \big(p(x),q(x)\big)-Laplacian-like operators of the following form:\begin{equation*}\displaystyle\left\{\begin{array}{ll}\displaystyle-\Delta^{l}_{p(x)}-\Delta^{l}_{q(x)}=\lambda g(x, u, \nabla u) & \mathrm{i}\mathrm{n}\ \Omega,\\\\u=۰ & \mathrm{o}\mathrm{n}\ \partial\Omega,\end{array}\right.\end{equation*}where \Delta^{l}_{r(x)} is the r(x)-Laplacian-like operators, \Omega is a smooth bounded domain in \mathbb{R}^{N}, \lambda is a real parameter and g is Carath\'eodory function satisfies the assumption of growth. The existence is proved by using Berkovits' topological degree.

Keywords:

Dirichlet problems , double phase problems , (p(x) , q(x))- Laplacian-like operators , topological degree methods