Polyharmonic Nonlinear Scalar Field Equations

TL;DR

This paper proves the existence of ground state solutions for polyharmonic nonlinear scalar field equations with subcritical growth, addressing analytical challenges of higher-order operators and establishing a new logarithmic Sobolev inequality.

Contribution

It extends existence results to higher-order polyharmonic equations and introduces a novel polyharmonic logarithmic Sobolev inequality.

Findings

Existence of ground state solutions for $(-\Delta)^m u=g(u)$

Development of a new polyharmonic logarithmic Sobolev inequality

Overcoming analytical challenges of higher-order operators

Abstract

In this paper, we present a result on the existence of ground state solutions for the polyharmonic nonlinear equation $(-\Delta)^m u=g(u)$, assuming that $g$ has a general subcritical growth at infinity, inspired by Berestycki and Lions \cite{BerestyckiLions}. In comparison with the biharmonic case studied in \cite{Med-Siem}, the presence of a higher-order operator gives rise to several analytical challenges, which are overcome in the present work. Furthermore, we establish a new polyharmonic logarithmic Sobolev inequality.