Electron. J. Differential Equations, Vol. 2017 (2017), No. 49, pp. 1-17.

Characterization of a homogeneous Orlicz space

Waldo Arriagada, Jorge Huentutripay

In this article we define and characterize the homogeneous Orlicz space $\mathcal{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ 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 $\mathcal{D}^{1,\Phi}_{\rm o}(\mathbb{R}^{N})$ 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.

Submitted August 31, 2016. Published February 16, 2017.
Math Subject Classifications: 46E30, 46T30, 35J20, 35J50.
Key Words: Homogeneous space; Orlicz space; eigenvalue problem; phi-Laplacian

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Waldo Arriagada
Department of Applied Mathematics and Sciences
Khalifa University, Al Zafranah, P.O. Box 127788
Abu Dhabi, United Arab Emirates
email: waldo.arriagada@kustar.ac.ae
Jorge Huentutripay
Instituto de Ciencias Físicas y Matemáticas
Universidad Austral de Chile
Casilla 567, Valdivia, Chile
email: jorge.huentutripay@uach.cl

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