Existence of solutions for a (p,q)-Laplace equation with Steklov boundary conditions

Here, the existence of at least one nontrivial solution for the (p,q)-Laplacian problem\[\left\{ \begin{array}{lc}div(|\nabla u|^{p-۲}\nabla u)+div(|\nabla u|^{q-۲}\nabla u)=f(x,u)\qquad &  x\in \Omega, \\\\|\nabla u|^{p-۲}\frac{\partial u}{\partial n}+ |\nabla u|^{q-۲}\frac{\partial u}{\partial n}=g(x,u)  & x\in  \partial  \Omega\end{array} \right.\]is done, where \Omega is a bounded domain in \mathbb{R}^N, N\geq ۳ and q, p\geq ۲, via variational methods.

Keywords:

(p , q)-Laplace equation , Steklov boundary conditions , Mountain Pass Theorem