Characterization of a homogeneous Orlicz space

Abstract

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.

Keywords

• Homogeneous space
• Orlicz space
• eigenvalue problem
• $\phi$-Laplacian