Ground State Solutions For Local-Nonlocal Shrodinger Equations In the Presence Of Two Critical Exponents

TL;DR

This paper investigates the existence and properties of ground state solutions for Schrödinger equations involving both local and nonlocal operators with two critical nonlinearities, addressing a novel and challenging PDE problem.

Contribution

It introduces a new approach based on a generalized Lieb translation theorem to handle the complex local-nonlocal critical nonlinearities, filling a gap in PDE theory.

Findings

Complete solution in the critical case for local and nonlocal settings.

Development of a generalized Lieb translation theorem for this context.

Discussion of positivity and regularity of ground states.

Abstract

In this paper, we address the existence of ground state solutions for Schrodinger equations in the presence of local and nonlocal operators and two critical nonlinearities associated with each operator. The situation is completely solved in the critical case both in the local and in the nonlocal settings. However, methods developed in these cases cannot extend to local-nonlocal operators when we have two critical power nonlinearities. This unprecedented situation in PDEs presents some challenges, and its resolution will open the door to solve similar problems. Our approach is essentially based on a subtle generalization of the Lieb translation theorem. We will also discuss the positivity of the ground states as well as their regularity.