| Summary: | In this article we define and characterize the homogeneous Orlicz space
$\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ where
$\Phi:\mathbb{R}\to [0,+\infty)$
is the N-function generated by an odd, increasing and not-necessarily
differentiable homeomorphism $\phi:\mathbb{R}\to\mathbb{R}$.
The properties of $\mathscr{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ are treated
in connection with the $\phi$-Laplacian eigenvalue problem
$$
-\hbox{div}\Big(\phi(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)
=\lambda\,g(\cdot)\phi(u)\quad\text{in }\mathbb{R}^N
$$
where $\lambda\in\mathbb{R}$ and $g:\mathbb{R}^N\to\mathbb{R}$ is measurable.
We use a classic Lagrange rule to prove that solutions of the $\phi$-Laplace
operator exist and are non-negative.
|
|---|
