A priori estimates of stable and finite Morse index solutions to elliptic equations that arise in Physics

TL;DR

This paper investigates the regularity, stability, and multiplicity of solutions to nonlinear elliptic equations from physics, revealing new insights into solution behavior and challenging existing conjectures through counterexamples and optimal regularity results.

Contribution

It constructs a counterexample showing bounded Morse index does not ensure regularity and establishes optimal regularity conditions for radial solutions of Hardy-Hénon equations.

Findings

Counterexample for singular behavior in dimensions 3-9

Optimal regularity results for Hardy-Hénon equations

Existence of multiple solutions under certain conditions

Abstract

This thesis studies qualitative properties of solutions to nonlinear elliptic equations of Poisson type with Dirichlet boundary conditions that arise from some physical phenomena, with a particular focus on regularity, stability, and multiplicity of solutions. Building on the modern framework of solution stability and Morse index theory, the work investigates how these notions influence regularity in nonlinear elliptic problems. A central contribution is the construction of a counterexample showing that bounded radial Morse index does not prevent singular behavior of solutions in dimensions three through nine, challenging a natural extension of the Brezis-V\'azquez regularity conjecture. In addition, optimal regularity results are established for radial solutions of a non-autonomous Hardy-H\'enon equation, identifying the precise range of dimensions for which regularity holds. The thesis also addresses existence and multiplicity results for elliptic equations involving nonlinearities with spatially vanishing coefficients. Under suitable assumptions, the existence of multiple distinct solutions is proved using variational and topological methods. Finally, the thesis outlines several directions for future research, including extensions of stability-based regularity techniques to non-autonomous problems and potential applications of these techniques to field theories arising in theoretical physics.