The Brezis-Nirenberg problem for mixed local and nonlocal operators

Abstract

In this note we present some existence results, in the spirit of the celebrated paper by Brezis and Nirenberg (CPAM, 1983), for a perturbed critical problem driven by a mixed local and nonlocal linear operator. We develop an existence theory, both in the case of linear and superlinear perturbations; moreover, in the particular case of linear perturbations we also investigate the mixed Sobolev inequality associated with this problem, detecting the optimal constant, which we show that is never achieved.

Keywords

• operators of mixed order
• sobolev inequality
• critical exponents
• existence theory