Biharmonic nonlinear scalar field equations

We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $${\mathrm{\Delta }}^{2}u=g(x,u)\text{in }{\mathbb{R}}^N$$ with a Carath'eodory function $g:$\mathbb{R}$^N\times $\mathbb{R}$\to $\mathbb{R}$$,$N\geq 5$. The regularity results give rise to the existence of ground state solutions provided that $g$ has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^2\log \mathrm{\mid }u\mathrm{\mid }dx\le \frac{N}{8}\log \left(C{\int }_{{\mathbb{R}}^N}\mathrm{\mid }\mathrm{\Delta }u{\mathrm{\mid }}^{2}dx\right),\text{for }u\in H^2({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}{u}^{2}dx=1,$$ for a constant $0<C< \big(\frac{2}{\pi e N}\big)^2$ and we characterize its minimizers.