Biharmonic nonlinear scalar field equations

Abstract

We prove a Brezis-Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $${\Delta }^{2}u=g(x,u)\text{in }{\mathbb{R}}^N$$ with a Carathéodory function $g:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$, $N\ge 5$. 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 $${\int }_{{\mathbb{R}}^N}|u{|}^2\log |u|dx\le \frac{N}{8}\log (C{\int }_{{\mathbb{R}}^N}|\Delta u{|}^{2}dx),\text{for }u\in H^2({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}{u}^{2}dx=1,$$ for a constant $0<C<(\frac{2}{\pi eN}{)}^2$ and we characterize its minimizers.