Infinitely many solutions for nonlinear superposition operators of mixed fractional order involving critical exponent

Abstract: This paper addresses elliptic problems involving the superposition of nonlinear fractional operators with the critical Sobolev exponent in sublinear regimes. We establish the existence of infinitely many nontrivial weak solutions using a variational framework that combines a truncation argument with the notion of genus. A central part of our analysis is the verification of the Palais--Smale $(\mathrm{PS})c$ condition for every $q \in (1, p{s_\sharp}^*)$, despite the challenges posed by the lack of compactness due to the critical exponent. On the one hand, our approach extends and generalizes earlier results by Garc{\'i}a Azorero and Peral Alonso \emph{[Trans. Amer. Math. Soc., 1991]} and by Da Silva, Fiscella, and Viloria \emph{[J. Differential Equations, 2024]}; on the other hand, the results obtained here complement the study of the Brezis--Nirenberg-type problem by Dipierro, Perera, Sportelli, and Valdinoci \emph{[Commun. Contemp. Math., 2024]} for $q = p$ and by Aikyn, Ghosh, Kumar, and Ruzhansky \emph{[arXiv:2504.05105, 2025]} for $p < q < p_{s_\sharp}^*$. Notably, our results are new even in the classical case $p = 2$, highlighting the broader applicability of the methods developed here.