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arXiv · 2507.12962

Polyharmonic Nonlinear Scalar Field Equations

Alessandro Cannone·Silvia Cingolani·Jarosław Mederski
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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.

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Alessandro Cannone, Silvia Cingolani, Jarosław Mederski. 2025-07-17. Polyharmonic Nonlinear Scalar Field Equations. https://arxiv.org/abs/2507.12962

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