The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field

Claudianor O. Alves; Rodrigo C. M. Nemer; Sergio H. Monari Soares

doi:10.3934/cpaa.2020276

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2021, Volume 20, Issue 1: 449-465. Doi: 10.3934/cpaa.2020276

This issue Previous Article Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food Next Article

The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field

1.
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande – PB, 58109-970, Brazil
2.
Instituto de Ciências Matemáticas e Computação, Universidade de São Paulo, São Carlos – SP, 13560-970, Brazil

^* Corresponding author
^* Corresponding author

Received: June 2020

Revised: September 2020

Early access: November 2020

Published: January 2021

The first author is supported by CNPq/Brazil

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Abstract

Nonlinear Schrödinger equations with an external magnetic field and a power nonlinearity with subcritical exponent $ p $ are considered. It is established a lower bound to the number of nontrivial solutions to these equations in terms of the topology of the domains in which the problem is given if $ p $ is suitably close to the critical exponent $ 2^* = 2N/(N-2) $, $ N \geq 3 $. To prove this lower bound, based on a proof of a result of Benci and Cerami, it is provided an abstract result that establishes Morse relations that are used to count solutions.

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Mathematics Subject Classification: Primary: 58E05, 35A15; Secondary: 35Q55, 35J15.

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