Abstract
In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality \(S\|u\|_{L^{p_*}(\partial\Omega)}^p \le \|u\|_{W^{1,p}(\Omega)}^p\) that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p * : = p(N − 1)/(N − p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.
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Fernández Bonder, J., Saintier, N. Estimates for the Sobolev trace constant with critical exponent and applications. Annali di Matematica 187, 683–704 (2008). https://doi.org/10.1007/s10231-007-0062-1
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DOI: https://doi.org/10.1007/s10231-007-0062-1