Global Compactness Result for a Brézis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator

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 $\Omega \subseteq {\mathbb{R}}^N$ is a smooth bounded domain with $N\ge 3$, $s\in (0,1),\lambda \in \mathbb{R}$ is a real parameter and $2^{\ast }=\frac{2N}{N-2}$ 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 $\lambda =0$.