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 smooth 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}. To the best of our knowledge, 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, proving the compactness result requires several delicate new and novel estimates, which we believe will be of independent interest for further studies of related problems. 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.