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

Abstract

The paper is concerned with Lane-Emden and Brezis-Nirenberg problems involving the affine $p$-laplace nonlocal operator ${\Delta }_p^{\mathcal{A}}$, which has been introduced in \cite{HJM5} driven by the affine $L^p$ energy ${\mathcal{E}}_{p,\Omega }$ 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 ${\mathcal{E}}_{p,\Omega }$ and by the comparison ${\mathcal{E}}_{p,\Omega }(u)\le \Vert u{\Vert }_{W_0^{1,p}(\Omega )}$ generally strict.

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