Singular solutions to $k$-Hessian equations with fast-growing nonlinearities

TL;DR

This paper investigates singular solutions to $k$-Hessian elliptic equations with rapidly growing nonlinearities, establishing existence, asymptotic behavior, and solution multiplicity on a unit ball.

Contribution

It introduces new results on the existence and asymptotics of singular solutions and explores the bifurcation structure and intersection properties of solutions.

Findings

Existence of radial singular solutions near the origin.

Asymptotic characterization of solutions' behavior.

Analysis of solution multiplicity and bifurcation depending on parameters.

Abstract

We study a class of elliptic problems, involving a $k$-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic behavior in a neighborhood of the origin. Furthermore, we study the multiplicity of regular solutions and bifurcation diagrams. An essential ingredient of this study is analyzing the number of intersection points between the singular and regular solutions for rescaled problems. In the particular case of the exponential nonlinearity, we obtain the convergence of regular solutions to the singular and analyze the intersection number depending on the parameter $k$ and the dimension $d$.