Award Date

5-1-2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Committee Member

Hossein Tehrani

Second Committee Member

David Costa

Third Committee Member

Zhonghai Ding

Fourth Committee Member

Pushkin Kachroo

Number of Pages

87

Abstract

Existing results provide the existence of positive solutions to a class of semilinear elliptic PDEs with logistic-type nonlinearities and harvesting terms both in RN and in bounded domains U ⊂ RN with N ≥ 3, when the carrying capacity of the environment is not constant. We consider these same equations in the exterior domain Ω, defined as the complement of the closed unit ball in RN , N ≥ 3, now with a Dirichlet boundary condition. We first show that the existing techniques forsolving these equations in the whole space RN can be applied to the exterior domain with some modifications. Then, as a second approach, we use the Kelvin transform to move the equation inside the unit ball, solve it there, using the techniques for bounded domains, and then re-apply the Kelvin transform to obtain a solution to the original equation. We are then confronted with the natural question of whether the two different approaches provide a multiplicity result for positive solutions in our exterior domain. As part of this work we prove a uniqueness result under further assumptions on the data. Finally, we briefly show that the Kelvin transform method can also be applied to the case of N = 2 with some slight adjustments, and that the solution obtained in this case also satisfies a similar uniqueness property.

Keywords

analysis; calculus of variations; differential equations; mathematics; nonlinear analysis; partial differential equations

Disciplines

Applied Mathematics | Mathematics

File Format

pdf

File Size

746 KB

Degree Grantor

University of Nevada, Las Vegas

Language

English

Rights

IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

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