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

TL;DR

This paper proves the existence of infinitely many solutions for a class of nonlinear elliptic problems involving mixed fractional operators with critical exponents, using variational methods and genus theory.

Contribution

It introduces a novel variational approach to handle critical Sobolev exponents in mixed fractional elliptic problems, establishing solutions even in the classical case.

Findings

Existence of infinitely many solutions established.

Verification of the Palais--Smale condition despite critical exponent challenges.

Results extend to classical cases, broadening applicability.

Abstract

This paper addresses a class of elliptic problems involving the superposition of nonlinear fractional operators with the critical Sobolev exponent in the sublinear regimes. We establish the existence of infinitely many nontrivial weak solutions using a variational framework combining a truncation argument with the notion of genus. A central part of our analysis is the verification of the Palais--Smale (PS) condition for the associated energy functional for every $q \in (1, p_{s_\sharp}^*)$, despite the challenges posed by the lack of compactness due to the critical exponent. The results obtained in the paper are new even in the classical case $p = 2$, highlighting the broader applicability of the methods developed here.