  In this article we study some optimal design problems related to nonstandard
growth eigenvalues ruled by the $g-$Laplacian operator. More precisely, given
$\Omega\subset \R^n$ and $\alpha,c>0$ we consider the optimization problem
$\inf \{ \lambda_\Omega(\alpha,E)\colon E\subset \Omega, |E|=c \}$, where
$\lambda_\Omega(\alpha,E)$ is related to the first eigenvalue to $$
-\text{div}(g( |\nabla u |)\tfrac{\nabla u}{|\nabla u|}) + g(u)\tfrac{u}{|u|}+
\alpha \chi_E g(u)\tfrac{u}{|u|} \quad \text{ in }\Omega $$ subject to
Dirichlet, Neumann or Steklov boundary conditions. \\ We analyze existence of
an optimal configuration, symmetry properties of them, and the asymptotic
behavior as $\alpha$ approaches $+\infty$.
