Global Compactness Result for a Brézis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator
Souptik Chakraborty, Diksha Gupta, Shammi Malhotra, Konijeti Sreenadh
TL;DR
The paper advances the understanding of compactness for critical Brezis–Nirenberg type problems involving a mixed local–nonlocal operator by establishing a global compactness (profile decomposition) result for Palais–Smale sequences. The decomposition identifies a base weak limit plus finitely many bubbles solving the limiting equation at infinity, with precise energy splitting and no interaction in the limit. This framework is then leveraged to prove a Coron-type existence result for high-energy positive solutions on annular domains when $\lambda=0$, valid in sufficiently high dimensions and under suitable geometric scaling of the domain. The work unifies local and nonlocal concentration phenomena, clarifies the role of the best Sobolev constant, and provides concrete criteria for the existence of high-energy solutions in topologically nontrivial settings.
Abstract
This paper investigates the profile decomposition of Palais-Smale sequences associated with a Brezis-Nirenberg type problem involving a combination of mixed local nonlocal operators, given by \begin{equation*} \left\{\begin{aligned} &-Δu + (-Δ)^s u - λu = |u|^{2^*-2}u \;\;\mbox{ in } Ω, &\quad u=0\,\mbox{ in }\mathbb{R}^N\setminus Ω. \end{aligned} \right. \end{equation*} where $Ω\subseteq \mathbb{R}^{N}$ is a smooth bounded domain with $N \geq 3$, $s\in (0,1),\,λ\in\mathbb{R}$ is a real parameter and $2^* = \frac{2N}{N - 2} $ denotes the critical Sobolev exponent. As an application of the derived global compactness result, we further study the existence of positive solution of the corresponding Coron-type problem (C. R. Acad. Sci. Paris Sér I Math, 299(7):209-212, 1984) when $λ=0$.