Bounded solutions of a $k$-Hessian equation in a ball

TL;DR

This paper investigates the existence and multiplicity of bounded, radially symmetric solutions to a $k$-Hessian equation in a ball, identifying critical exponents and explicitly expressing solutions using Bliss functions.

Contribution

It introduces a new Emden-Fowler transformation for the $k$-Hessian operator and determines solution multiplicity based on critical exponents, including explicit solutions in the critical case.

Findings

Exactly one or two solutions in the critical case depending on parameters.

Identification of a new critical exponent $q_{JL}(k)$ for solution multiplicity.

Explicit solutions expressed via Bliss functions in the critical case.

Abstract

We consider the problem \begin{equation}\label{Eq:Abstract} (1)\;\;\;\begin{cases} S_k(D^2u)= \lambda (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes the unit ball in $\mathbb{R}^n$, $n>2k$ ($k\in \mathbb{N}$), $\lambda>0$ and $q > k$. We study the existence of negative bounded radially symmetric solutions of (1). In the critical case, that is when $q$ equals Tso's critical exponent $q=\frac{(n+2)k}{n-2k}=:q^*(k)$, we obtain exactly either one or two solutions depending on the parameters. Further, we express such solutions explicitly in terms of Bliss functions. The supercritical case is analysed following the ideas develop by Joseph and Lundgren in their classical work [27]. In particular, we establish an Emden-Fowler transformation which seems to be new in the context of the $k$-Hessian operator. We also find a critical exponent, defined by \begin{equation*} q_{JL}(k)= \begin{cases} k\frac{(k+1)n-2(k-1)-2\sqrt{2[(k+1)n-2k]}}{(k+1)n-2k(k+3)-2\sqrt{2[(k+1)n-2k]}}, & n>2k+8,\\ \infty, & 2k < n \leq 2k+8, \end{cases} \end{equation*} which allows us to determinate the multiplicity of the solutions to (1) int the two cases $q^*(k)\leq q < q_{JL}(k)$ and $q\geq q_{JL}(k)$. Moreover, we point out that, for $k=1$, the exponent $q_{JL}(k)$ coincides with the classical Joseph-Lundgren exponent.