An Interpolation of Hardy Inequality and Trudinger-Moser Inequality in RN and Its Applications

We establish an interpolation of Hardy inequality and Trudinger-Moser inequality in R-N (N >= 2). Denote parallel to u parallel to(1,tau) = (integral(RN) (vertical bar del u vertical bar(N) + tau vertical bar u vertical bar(N))dx)(1/N) for any tau > 0. There holds sup(parallel to u parallel to 1,tau <= 1) integral(RN) 1/vertical bar x vertical bar(beta) {e(alpha vertical bar u vertical bar N/(N-1)) - Sigma(N-2)(m=0) alpha(m)vertical bar u vertical bar(mN/(N-1))/m!} dx < infinity if and only if alpha/alpha(N) + beta/N <= 1, where 0 <= beta < N, alpha(N) = N omega(1/(N-1))(N-1), omega(N-1) is the volume of the unit sphere SN-1. The above interpolation can be used to establish sufficient conditions under which the quasilinear nonhomogeneous partial differential equation -Delta(N)u + V(x)vertical bar u vertical bar(N-2)u = f(x,u)/vertical bar x vertical bar(beta) + epsilon h(x) in R-N has weak solutions, where -Delta(N)u = -div(vertical bar del u vertical bar(N-2)del u) is the N-Laplacian, V is a continuous potential, f behaves like e(alpha vertical bar u vertical bar N/(N-1)) when vertical bar u vertical bar -> infinity, h is an element of (W-1,W-N(R-N))*, 0 < beta < N, and epsilon > 0.