Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case

TL;DR

This paper derives an explicit formula for the critical mass and proves the existence and characterization of energy ground state solutions for a mixed local-nonlocal Schrödinger equation at critical exponents, advancing understanding of such equations.

Contribution

It establishes an explicit critical mass formula and proves the existence and optimality of ground state solutions for the mixed local-nonlocal Schrödinger equation.

Findings

Explicit critical mass formula derived

Existence of optimizer for Gagliardo-Nirenberg inequality proved

Energy ground state solutions characterized as optimizers

Abstract

Our first main contribution consists in establishing an explicit formula of the critical mass via the best constant of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal Laplacian. We also prove the existence of an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator. We then show that the optimizer (after some suitable scaling) is an energy ground state solution (with critical mass ). This is a key ingredient to determine sufficient and necessary conditions of existence and non-existence of energy ground state solutions in the in-between critical exponents case. Finally, we show that the energy ground state solution uc0 is an optimizer of the Gagliardo-Nirenberg inequality for the mixed local-nonlocal operator.