Sharp Existence of Ground States Solutions for a Class of Elliptic Equations with Mixed Local and Nonlocal Operators and General Nonlinearity

In this paper, we study the existence/non-existence of ground states for the following type of elliptic equations with mixed local and nonlocal operators and general nonlinearity: (−▵)su−▵u+λu=f(u),x∈RN, which is driven by the superposition of Brownian and Lévy processes. By considering a constrained variational problem, under suitable assumptions on f, we manage to establish a sharp existence of the ground state solutions to the equation considered. These results improve the ones in the existing reference.