Dissertations, Theses, and Capstone Projects

Date of Degree

6-2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy

Program

Mathematics

Advisor

Marcello Lucia

Committee Members

Yi Li

Enrique Pujals

Subject Categories

Analysis

Keywords

semilinear elliptic equations, Dirac measure data, variational sub- and supersolutions, exponential nonlinearity, Trudinger–Moser inequality, vortex condensation

Abstract

This dissertation utilizes a variational framework for semilinear elliptic equations in two dimensions with Dirac measure data. The central objects of study are equations of the form −ΔU = f(U) + Σj=1N αjδpj on bounded Lipschitz domains Ω ⊂ ℝ² with homogeneous Dirichlet boundary condition, and on a flat torus 𝕋², where αj is positive for the Dirichlet setting and αj is negative on the torus. Solutions are obtained by minimizing the restriction of an energy functional to an order interval determined by explicit sub- and supersolutions; the Euler–Lagrange equation is recovered through a truncation argument that requires no monotonicity of f, and no iterative schemes.

Chapter 2 develops the variational sub-/supersolution method for both scalar equations and coupled systems, relaxing the uniform L∞ bounds of Struwe's classical formulation to Ls domination (s > 1) on the order interval. Chapter 3 applies this framework to the singular Dirichlet problem with nonlinearities exhibiting at most exponential growth of the type f(t) ~ J eβt with J, β > 0. In the case of one singularity we establish a sharp dichotomy at the critical threshold α1β = 4π: below it, weak solutions exist for sufficiently small coupling constant J; at or above it, no weak solution exists. This extends the admissible parameter range from α1β ≤ 2π in a work of Dhanya, Giacomoni, and Prashanth to the full subcritical range α1β < 4π. Eigenvalue obstructions provide additional nonexistence criteria. Motivated by some vortex condensation model on 𝕋², Chapter 4 extends the results obtained by Caffarelli–Yang to general sublinear nonlinearities and singular background data. Existence is established via a different supersolution construction than the one used in the Dirichlet setting, which is necessary due to the opposite sign of the Dirac sources and sublinear growth assumption on f. Chapter 5 extends the analysis of the Dirichlet setting of Chapter 3 to coupled two-field Dirichlet systems which have a variational structure, a setting where monotone iteration generically fails. Both diagonal and off-diagonal eigenvalue obstructions are derived; the off-diagonal obstruction, which detects nonexistence driven by cross-field interaction, has no scalar analogue. The theory is then applied to an exponential system.

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