Biharmonic nonlinear scalar field equations

We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear
equation $$\Delta^2u=g(x,u)\qquad\text{ in }\mathbb{R}^N$$ with a Carathéodory function $g:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$, $N\ge5$. 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 newbiharmonic logarithmic Sobolev inequality
$$\int_{\mathbb{R}^N}|u|^2\log|u|\,dx \le \frac{N}{8}\log\left(C\int_{\mathbb{R}^N}|\Delta u|^2\,dx\right), \quad\text{ for } u\in H^2(\mathbb{R}^N), \int_{\mathbb{R}^N}u^2\,dx=1,$$
for a constant $0<C<\left(\frac{2}{\pi e N}\right)^2$ and we characterize its minimizers.