Estimates for the Sobolev trace constant with critical exponent and applications

Abstract

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality S∥u∥Lp*(∂Ω)p≤∥u∥W1,p(Ω) 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. © 2007 Springer-Verlag.