Normalized solutions for a class of scalar field equations involving mixed fractional Laplacians

The purpose of this article is to establish sharp conditions for the existence of normalized solutions to a class of scalar field equations involving mixed fractional Laplacians with different orders. This study includes the case when one operator is local and the other one is non-local. This type of equation arises in various fields ranging from biophysics to population dynamics. Due to the importance of these applications, this topic has very recently received an increasing interest. In this article, we provide a complete description of the existence/non-existence of ground state solutions using constrained variational approaches. This study addresses the mass subcritical, critical and supercritical cases. Our model presents some difficulties due to the "conflict" between the different orders and requires a novel analysis, especially in the mass supercritical case. We believe that our results will open the door to other valuable contributions in this important field.