The best constant in the G-N inequality for the mixed local and Nonlocal Laplacian

TL;DR

This paper determines the optimal constant in a G-N inequality involving mixed local and nonlocal Laplacians, introducing a novel method to handle the complex operator and identify the optimizer as the ground state solution.

Contribution

The paper develops an innovative approach to find the best constant in the G-N inequality for mixed operators, overcoming regularity challenges.

Findings

Established the best constant in the G-N inequality for mixed Laplacians.

Proved the optimizer is the ground state solution of the associated equation.

Developed a new method applicable to complex operators with challenging regularity.

Abstract

In this paper, we establish the best constant in the G-N inequality for the mixed local and nonlocal Laplacian. In our problem, classical methods cannot apply directly since regularity results for the operator under study seem to be highly challenging. We build an innovative method that not only enabled us to prove that the optimizer of the best constant is the ground state solution of the equation, but also to establish the best constant.