THE BEST CONSTANT FOR L∞-TYPE GAGLIARDO-NIRENBERG INEQUALITIES

In this paper we derive the best constant for the following L-infinity-typ e Gagliardo-Nirenberg interpolation inequality parallel to u parallel to(L infinity) <= C-q,C-infinity,C-p parallel to u parallel to(1-theta)(Lq+1) parallel to del u parallel to(theta)(Lp), theta = pd/dp + (p - d)(q + 1), where parameters q and p satisfy the conditions p > d >= 1, q >= 0. The best constant C-q,C-infinity,C-p is given by C-q,C-infinity,C-p = theta(-theta/p) (1 - theta)(theta/p) M-c(-theta/d) , M-c := integral(Rd) u(c,infinity)(q+1)dx, where u(c,infinity) is the unique radial non-increasing solution to a generalized Lane-Emden equation. The case of equality holds when u = Au-c,Au-infinity(lambda(x - x(0))) for any real numbers A, lambda > 0 and x(0) is an element of R-d. In fact, the generalized Lane-Emden equation in R-d contains a delta function as a source and it is a Thomas-Fermi type equation. For q = 0 or d = 1, u(c,infinity) have closed form solutions expressed in terms of the incomplete Beta functions. Moreover, we show that u(c,m) -> u(c,infinity) and C-q,C-m,C-p -> C-q,C-infinity,C-p as m -> +infinity for d = 1, where u(c,m) and C-q,C-m,C-p are the function achieving equality and the best constant of L-m-type Gagliardo-Nirenberg interpolation inequality, respectively.