arXiv subjects
Publications and source records attributed to Mohamed Khalil Zghal.
We establish a sharp Adams-type inequality invoking a Hardy inequality for any even dimension. This leads to a non compact Sobolev embedding in some Orlicz space. We also give a description of the lack of compactness of this embedding in the spirit of \cite{Bahouri}.
In this paper, we investigate the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ into the Orlicz space $L^{ϕ_p}(\R^2)$ associated to the function $ϕ_p$ defined by $ϕ_p(s):={\rm{e}^{s^2}}-\Sum_{k=0}^{p-1} \frac{s^{2k}}{k!}\cdot$ We also undertake the study of a nonlinear wave equation with exponential growth where the Orlicz norm $\|.\|_{L^{ϕ_p}}$ plays a crucial role. This study includes issues of global existence, scattering and qualitative study.
This paper is devoted to the description of the lack of compactness of the Sobolev space $H^2_{rad}(\R^4)$ in the Orlicz space $\mathcal{L}(\R^4)$. The approach that we adopt to establish this characterization is in the spirit of the one adopted in the case of $H^1_{rad}(\R^2)$ into the Orlicz space $\mathcal{L}(\R^2)$ in \cite{Bahouri}.