Biharmonic nonlinear scalar field equations

TL;DR

This paper establishes regularity results and existence of ground state solutions for biharmonic nonlinear equations in high dimensions, introduces a new logarithmic Sobolev inequality for biharmonic operators, and characterizes its minimizers.

Contribution

It proves a regularity theorem for solutions to biharmonic equations, demonstrates existence of ground states under subcritical growth, and formulates a novel biharmonic logarithmic Sobolev inequality with minimizer characterization.

Findings

Regularity results for biharmonic equations with nonlinearities.

Existence of ground state solutions under subcritical growth.

A new biharmonic logarithmic Sobolev inequality and its minimizers.

Abstract

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