Critical and Supercritical Adams' Inequalities with Logarithmic Weights

In this paper, we obtain several Adams-type inequalities with logarithmic weights on the unit ball B in R-n, where n >= 3. Firstly, we show that for any beta is an element of (0, 1), the following critical Adams' inequality sup(u is an element of W0,r2,n/2 (B,omega), parallel to u parallel to omega <= 1) integral(B) (exp) (alpha vertical bar u vertical bar n/(n-2)(1-beta) dx < infinity holds if and only if alpha <= alpha(beta) = n [(n - 2)(n/2) nV(n)](2/(n-2)(1-beta)) (1 - beta)(1/1-beta), where V-n = pi n/2/Gamma(n/2 +1) is the volume of the unit ball B in R-n, W-0,r(2,n/2) (B, omega) is the radial weighted Sobolev space under the norm parallel to u parallel to(omega) = (integral(B)vertical bar Delta u vertical bar(n/2) omega (x) dx)(2/n) with omega(x) = (log 1/vertical bar x vertical bar)(beta(n/2 -1)) or omega(x) = (log e/vertical bar x vertical bar)(beta(n/2 -1)). Secondly, we prove the following two supercritical Adams' inequalities sup(u is an element of W0,r2,n/2 (B,omega), parallel to u parallel to omega <= 1) integral(B) exp (alpha(beta) + vertical bar x vertical bar(m))vertical bar u vertical bar n/(n-2)(1-beta) dx < infinity and sup(u is an element of W0,r2,n/2 (B,omega), parallel to u parallel to omega <= 1) integral(B) exp (alpha(beta) + vertical bar u vertical bar n/(n-2)(1-beta)(+vertical bar x vertical bar m))dx < infinity, where m is some positive number. Moreover, we prove the existence of extremals for these Adams-type inequalities.