Biharmonic Nonlinear Scalar Field Equations

We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation Delta(2)u = g(x, u) in R-N with a Caratheodory function g : R-N x R -> R, N >= 5. The regularity results give rise to the existence of ground state solutions provided that g has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality integral(RN) vertical bar u vertical bar(2) log vertical bar u vertical bar dx <= N/8 log (C integral(RN)vertical bar Delta u vertical bar(2) dx), for u is an element of H-2(R-N), integral(RN) u(2) dx = 1, for a constant 0 < C < (2/pi eN)(2) and we characterize its minimizers.