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

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.