Global Compactness Result for a Br\'ezis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator

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} &-\Delta u + (-\Delta)^s u - \lambda u = |u|^{2^*-2}u \;\;\mbox{ in } \Omega, &\quad u=0\,\mbox{ in }\mathbb{R}^N\setminus \Omega. \end{aligned} \right. \end{equation*} where $\Omega\subseteq \mathbb{R}^{N}$ is a smooth bounded domain with $N \geq 3$, $s\in (0,1),\,\lambda\in\mathbb{R}$ 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 S\'{e}r I Math, 299(7):209-212, 1984) when $\lambda=0$.