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On the Closing Lemma problem for the torus
Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension
| 1. | Department of Mathematics, Wayne State University, Detroit, MI 48202, United States |
| 2. | Department of Mathematics, Information School, Renmin University of China, Beijing 100872, China |
$\lambda_p(\Omega)=$i n f$_u\in H_0^1(\Omega),$u≠0(for some u)$\|\|\nabla u\|\|_2^2/\|\|u\|\|_p^2,$
where $|\|\cdot\||_p$ denotes $L^p$ norm. We derive in this paper a sharp form of the following improved Moser-Trudinger inequality involving the $L^p$-norm using the method of blow-up analysis:
$s u p_{u\in H_0^1(\Omega),\|\|\nabla u\|\|_2=1}\int_{\Omega} e^{4\pi (1+\alpha\|\|u\|\|_p^2)u^2}dx<+\infty$
for $0\leq \alpha <\lambda_p(\Omega)$, and the supremum is infinity for all $\alpha\geq \lambda_p(\Omega)$. We also prove the existence of the extremal functions for this inequality when $\alpha$ is sufficiently small.
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