Differential and Integral Equations

Description of the lack of compactness in Orlicz spaces and applications

Ines Ben Ayed and Mohamed Khalil Zghal

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In this paper, we investigate the lack of compactness of the Sobolev embedding of $H^1(\mathbb R^2)$ into the Orlicz space $L^{{\phi}_p}(\mathbb R^2)$ associated to the function $\phi_p$ defined by $ \phi_p(s):={\rm{e}^{s^2}}-\sum_{k=0}^{p-1} \frac{s^{2k}}{k!}$. We also undertake the study of a nonlinear wave equation with exponential growth where the Orlicz norm $\|.\|_{L^{\phi_p}}$ plays a crucial role. This study includes issues of global existence, scattering and qualitative study.

Article information

Differential Integral Equations, Volume 28, Number 5/6 (2015), 553-580.

First available in Project Euclid: 30 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B33: Critical exponents 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Ben Ayed, Ines; Zghal, Mohamed Khalil. Description of the lack of compactness in Orlicz spaces and applications. Differential Integral Equations 28 (2015), no. 5/6, 553--580. https://projecteuclid.org/euclid.die/1427744101

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