Multiplicity Results of Solutions to Non-Local Magnetic Schrödinger–Kirchhoff Type Equations in RN

Kisoeb Park

doi:10.3390/axioms11020038

Abstract

In this paper, we establish the existence of a nontrivial weak solution to Schrödinger-kirchhoff type equations with the fractional magnetic field without Ambrosetti and Rabinowitz condition using mountain pass theorem under a suitable assumption of the external force. Furthermore, we prove the existence of infinitely many large- or small-energy solutions to this problem with Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem by applying the variational methods, that is, the fountain and the dual fountain theorem with Cerami condition.

Highlights

The Schrödinger equation plays the role of Newton’s laws and conservation of energy in classical mechanics
The significant development of the Bose-Einstein condensates revived researches regarding the nonlinear waveforms for the nonlinear Schrödinger equations with external potentials and the related nonlinear partial differential equations
Axioms 2022, 11, 38 is to approach the problem by applying the variational methods, namely, the fountain and the dual fountain theorem with Cerami condition

Summary

Introduction

The Schrödinger equation plays the role of Newton’s laws and conservation of energy in classical mechanics. The present paper is devoted to the existence of solutions for the following Kirchhoff type equation with the fractional magnetic field A function u ∈ HsA,V (RN, C) is called weak solution of problem (3) if u satisfies

Results

Conclusion