论文标题
对于$ k $ -Hessian方程的解决方案的存在，多重性和分类结果
Existence, multiplicity and classification results for solutions to $k$-Hessian equations with general weights
论文作者
论文摘要
本文的目的是研究$ k $ -Hessian方程的负面经典解决方案，涉及具有一般重量的非线性\ begin {equination} \ label {eq：ma：0} \ tag {$ p $} \ begin {case {case} s_k（d^2u^2u^2u^2u） b，\\ u = 0＆\ mbox {on} \部分b。功能$rρ'（r）/ρ（r）$满足径向方向上非常通用的条件$ r = | x | $。我们显示了解决问题的解决方案的存在，不存在和多重性\ eqref {eq：ma：0}。用于证明的主要技术是与与\ eqref {eq：ma：0}方程相关的非自治动力系统相关的相平面分析。此外，使用上述非自治系统，我们给出$ p_2 $ - ，$ p_3^+$ - ，$ p_4^+$ - ，$ p_4^+$ - 解决方案的解决方案\ begin \ begin {equation*} \ begin {case {case {case {case} s_k（case} s_k（d^2 w） \ end {equation*}在整个空间上给出的$ \ mathbb r^n \！$。特别是，我们描述了新的解决方案类别：快速衰减$ p^+_ 3 $ -Solutions和$ P_4^+$ - 解决方案。
The aim of this paper is to study negative classical solutions to a $k$-Hessian equation involving a nonlinearity with a general weight \begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= λρ(|x|) (1-u)^q &\mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B. \end{cases} \end{equation} Here, $B$ denotes the unit ball in $\mathbb R^n\!$, $n>2k$, $λ$ is a positive parameter and $q>k$ with $k\in \mathbb N$. The function $rρ'(r)/ρ(r)$ satisfies very general conditions in the radial direction $r=|x|$. We show the existence, nonexistence, and multiplicity of solutions to Problem \eqref{Eq:Ma:0}. The main technique used for the proofs is a phase-plane analysis related to a non-autonomous dynamical system associated to the equation in \eqref{Eq:Ma:0}. Further, using the aforementioned non-autonomous system, we give a comprehensive characterization of $P_2$-, $P_3^+$-, $P_4^+$-solutions to the related problem \begin{equation*} \begin{cases} S_k(D^2 w)= ρ(|x|) (-w)^q, \\ w<0, \end{cases} \end{equation*} given on the entire space $\mathbb R^n\!$. In particular, we describe new classes of solutions: fast decay $P^+_3$-solutions and $P_4^+$-solutions.
