arXiv subjects

Justino Sánchez

Publications and source records attributed to Justino Sánchez.

6 recordsLinked to original sources

arXiv2025

A k-Hessian equation with a power nonlinearity source and self-similarity

We study existence and uniqueness of spherically symmetric solutions of S_k(D^2v)+beta xi\cdot\nabla v+αv+\abs{v}^{q-1}v=0 in R^n, where α,βare real parameters, n>2,\, q>k\geq 1 and S_k(D^2v) stands for the k-Hessian operator of v. Our results are based mainly on the analysis of an associated dynamical system and energy methods. We derive some properties of the solutions of the above equation for different ranges of the parameters αand β. In particular, we describe with precision its asymptotic behavior at infinity. Further, according to the position of q with respect to the first critical exponent \frac{(n+2)k}{n} and the Tso critical exponent \frac{(n+2)k}{n-2k} we study the existence of three classes of solutions: crossing, slow decay or fast decay solutions. In particular, if k>1 all the fast decay solutions have a compact support in R^n. The results also apply to construct self-similar solutions of type I to a related nonlinear evolution equation. These are self-similar functions of the form u(t,x)=t^{-α}v(xt^{-β}) with suitable αand β.

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arXiv2023

Existence of solutions for a $k$-Hessian equation and its connection with self-similar solutions

Let $α,β$ be real parameters and let $a>0$. We study radially symmetric solutions of \begin{equation*} S_k(D^2v)+αv+βξ\cdot\nabla v=0,\, v>0\;\; \mbox{in}\;\; \mathbb{R}^n,\; v(0)=a, \end{equation*} where $S_k(D^2v)$ denotes the $k$-Hessian operator of $v$. For $α\leq\frac{β(n-2k)}{k}\;\;\mbox{and}\;\;β>0$, we prove the existence of a unique solution to this problem, without using the phase plane method. We also prove existence and properties of the solutions of the above equation for other ranges of the parameters $α$ and $β$. These results are then applied to construct different types of explicit solutions, in self-similar forms, to a related evolution equation. In particular, for the heat equation, we have found a new family of self-similar solutions of type II which blows up in finite time. These solutions are represented as a power series, called the Kummer function.

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arXiv2022

Existence, multiplicity and classification results for solutions to $k$-Hessian equations with general weights

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.

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arXiv2020

A characterization of semistable radial solutions of k-Hessian equations

We characterize semistable radial solutions of the equation $S_k\left(D^2u\right)=g(u)\;\mbox{in } B_1$, where $B_1$ is the unit ball of $\mathbb{R}^n$, $D^2u$ is the Hessian matrix of $u,\,g$ is a positive $C^1$ nonlinearity and $S_k\left(D^2u\right)$ denotes the $k$-Hessian operator of $u$. This class of radial solutions has been recently introduced by the authors in [8]. The proofs are new relative to those given in [8] and focus on the structure of the equation directly, thereby improving some previous results.

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arXiv2018

Asymptotic behavior of solutions of a k-Hessian evolution equation

We study the long-time behavior of solutions of the $k$-Hessian evolution equation $u_t=S_{k}(D^2 u)$, posed on a bounded domain of the $n$-dimensional space with homogeneous boundary conditions. To this end, we construct a separable solution and we show that the long-time behavior of $u$ is precisely described by this special solution. Further, we initiate the study of that dynamic phenomenon on the entire space, providing a new class of explicit and radially symmetric self-similar solutions that we call $k$-Barenblatt solutions. These solutions present some common properties as those of well-known Barenblatt solutions for the porous media equation and the $p$-Laplacian equation. It is known that self-similar solutions are important in describing the intermediate asymptotic behavior of general solutions.

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arXiv2015

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

We consider the problem \begin{equation}\label{Eq:Abstract} (1)\;\;\;\begin{cases} S_k(D^2u)= λ(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 $\mathbb{R}^n$, $n>2k$ ($k\in \mathbb{N}$), $λ>0$ 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.

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