Biharmonic Nonlinear Scalar Field Equations

Abstract

Abstract We prove a Brezis–Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ \begin{align*} & \Delta^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N\end{align*}$$with a Carathéodory 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 $$ \begin{align*} & \int_{\mathbb{R}^N}|u|^2\log |u|\, \text{d}x\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|\Delta u|^2\, \text{d}x \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\mathbb{R}^N}u^2\, \text{d}x = 1, \end{align*}$$for a constant $0<C< \big (\frac {2}{\pi e N}\big )^2$ and we characterize its minimizers.