Asymptotic behaviour of solutions to the anisotropic doubly critical equation

TL;DR

This paper analyzes the asymptotic behavior of solutions to a complex anisotropic doubly critical PDE, revealing that solutions behave similarly to Euclidean cases and introducing novel techniques even in classical settings.

Contribution

It provides a comprehensive asymptotic analysis of solutions to the anisotropic doubly critical equation, with new methods applicable even in Euclidean contexts.

Findings

Solutions exhibit Euclidean-like asymptotic features near zero and infinity.

New analytical techniques are developed that are applicable beyond the anisotropic setting.

The study extends understanding of critical PDEs with anisotropic operators.

Abstract

The aim of this paper is to deal with the anisotropic doubly critical equation $$-\Delta_p^H u - \frac{\gamma}{[H^\circ(x)]^p} u^{p-1} = u^{p^*-1} \qquad \text{in } \R^N,$$ where $H$ is in some cases called Finsler norm, $H^\circ$ is the dual norm, $1<p<N$, $0 \leq \gamma< \left((N-p)/p\right)^p$ and $p^*=Np/(N-p)$. In particular, we provide a complete asymptotic analysis of $u \in \mathcal{D}^{1,p}(\R^N)$ near the origin and at infinity, showing that this solution has the same features of its euclidean counterpart. Some of the techniques used in the proofs are new even in the Euclidean framework.