A nonstandard growth Steklov optimization problem with volume constraint

In this article we study an optimal design problem for a nonstandard growth Steklov eigenvalues ruled by the $g-$Laplacian operator. More precisely, given $\Omega\subset \mathbb{R}^n$ and $\alpha,c>0$ we analyze existence and symmetry properties of solution of the optimization problem $\inf \{ \lambda(\alpha,E)\colon E\subset \Omega, |E|=c \}$, where, for a suitable function $u(\alpha,E)$, $\lambda(\alpha,E)$ solves \begin{equation*}\begin{cases} -÷(g( |\nabla u |)\frac{\nabla u}{|\nabla u|}) + (1+\alpha \chi_E)g( |\nabla u |)\frac{\nabla u}{|\nabla u|} =0& \text{ in } \Omega,\\ g(|\nabla u|)\frac{\nabla u}{|\nabla u|} \cdot \eta = \lambda g(|u|)\frac{u}{|u|} &\text{ on } \partial\Omega \end{cases} \end{equation*} being $g$ the derivative of a Young function, and $\eta$ the unit outward normal derivative. We analyze the behavior of the optimization problem as $\alpha$ approaches infinity and its connection of the trace embedding for Orlicz-Sobolev functions.