Sharp Sobolev inequalities with interior norms

In this paper, we establish some new sharp Sobolev inequalities on any smoo th bounded domain Omega subset of R-n. Let S-1 and S be the sharp constants corresponding to the Sobolev embedding and trace inequalities respectively . If n greater than or equal to 4, there exist constants A(Omega), A(1)(Ome ga) > 0 such that For All u is an element of H-1(Omega) \\u\\(2n/(n-2), Omega)(2) less than or equal to 2(2/n)S(1)\\del u\\(2)(2, O mega) + A(Omega)\\u\\(2n/(n-1), Omega)(2) and \\u\\(2(n-1)/(n-2), partial derivative Omega)(2) less than or equal to S \\ del u\\(2)(2, Omega) + A(1)(Omega)\\u\\(2n/(n-1), Omega)(2); If n = 3, for any k(3) > 3, there exist constants A(Omega, k(3)), A(1)(Omeg a, k(3)) > 0 such that For All u is an element of H-1(Omega) \\u\\<INF>2n/(n -2</INF>,) <INF>Omega</INF><SUP>2</SUP> less than or equal to 2<SUP>2/n</SUP>S<INF>1</INF> . \\del u\\<INF>2, Omega</INF><SUP>2</SUP> + A(Omega, k<INF>3</INF>) \\u\\<INF>k3, Omega</INF><SUP>2 </SUP>and \\u\\(2(n-1)/(n-2), partial derivative Omega)(2) less than or equal to S\\d el u\\(2)(2, Omega) + A(1)(Omega, k(3))\\u\\(2)(k3, Omega).( )