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- W4283661962 abstract "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)= lambda rho(|x|) (1-u)^q &mbox{in };; B, u=0 &mbox{on }partial B. end{cases} end{equation} Here, $B$ denotes the unit ball in $mathbb R^n!$, $n>2k$, $lambda$ is a positive parameter and $q>k$ with $kin mathbb N$. The function $rrho'(r)/rho(r)$ 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)= rho(|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." @default.
- W4283661962 created "2022-06-29" @default.
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- W4283661962 date "2022-06-23" @default.
- W4283661962 modified "2023-09-24" @default.
- W4283661962 title "Existence, multiplicity and classification results for solutions to $k$-Hessian equations with general weights" @default.
- W4283661962 doi "https://doi.org/10.48550/arxiv.2206.11942" @default.
- W4283661962 hasPublicationYear "2022" @default.
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