Least energy solutions for affine $p$-Laplace equations involving subcritical and critical nonlinearities

TL;DR

This paper investigates the existence of least energy positive solutions for affine p-Laplace equations with subcritical and critical nonlinearities, addressing challenges due to non-convexity of the associated energy functional.

Contribution

It introduces new results on existence and nonexistence of solutions for affine p-Laplace equations involving nonlocal operators and nonlinearities, expanding understanding of these complex problems.

Findings

Existence of positive solutions under certain conditions.

Nonexistence results in critical cases.

Analysis of the affine p-Laplace operator's properties.

Abstract

The paper is concerned with Lane-Emden and Brezis-Nirenberg problems involving the affine $p$-laplace nonlocal operator $\Delta_p^{\cal A}$, which has been introduced in \cite{HJM5} driven by the affine $L^p$ energy ${\cal E}_{p,\Omega}$ from convex geometry due to Lutwak, Yang and Zhang \cite{LYZ2}. We are particularly interested in the existence and nonexistence of positive $C^1$ solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ${\cal E}_{p,\Omega}$ and by the comparison ${\cal E}_{p,\Omega}(u) \leq \Vert u \Vert_{W^{1,p}_0(\Omega)}$ generally strict.