Brezis-Nirenberg problems for mixed local-nonlocal operators with superlinear perturbations: compactness and applications

In this paper, we consider the following mixed local nonlocal Brezis-Nirenberg problem \begin{equation}\label{crit_pro_abstract}\tag{$\mathcal{P}_{2^*}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{2^*-2}u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^N \setminus \Omega, \end{equation} where $\Omega\subset\mathbb{R}^N$ is a bounded domain, $N\geq3$, $s\in(0,1)$, $\lambda>0$, and $2\leq p<2^*=\frac{2N}{N-2}$. We establish a compactness result for the following class of subcritical/critical problems \begin{equation}\label{sub_pro_abstract}\tag{$\mathcal{P}_{p_n}$} -\Delta u+(-\Delta)^s u=\lambda |u|^{p-2}u+|u|^{p_n-2}u\text{ in }\Omega,\quad u=0\text{ in }\mathbb{R}^N \setminus \Omega, \end{equation} where $p_n \in (p,2^* ]$ and $p_n\to 2^*$. Specifically, for $p \in (2+\frac{4s}{N-2},2^*)$ when $N>6-4s$, and for $p \in (2^*-1,2^*)$ when $N\leq6-4s$, we prove that any bounded sequence of solutions $\{u_n\}$ to \eqref{sub_pro_abstract} is relatively compact in the energy space, and converges strongly to a nontrivial solution to \eqref{crit_pro_abstract}. This is the first paper to address this type of compactness result for a non-homogeneous operator. Due to the presence of the non-homogeneous operator, our proof requires a non-trivial adaptation of the methods developed by Devillanova and Solimini (Adv. Differential Equations, 2002) and Yan, Yang, and Yu (J. Funct. Anal., 2015). As an application of this compactness result, under the same ranges of $N$ and $p$, we prove that \eqref{crit_pro_abstract} admits infinitely many sign-changing solutions. We anticipate that our methodology will be applicable to a broader class of related problems.