Polyharmonic Nonlinear Scalar Field Equations

In this paper, we present a result on the existence of ground state solutions for the polyharmonic nonlinear equation $(-\mathrm{\Delta }{)}^{m}u=g(u)$, 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.