A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.&nbsp

Existence of radial solutions for k-Hessian system

In this paper, we consider the existence of radial solutions to a $k$-Hessian system in a general form. The existence of radial solutions is obtained under the assumptions that the nonlinearities in the given system satisfy $k$-superlinear, $k$-sublinear or $k$-asymptotically linear at the origin and infinity, respectively. The results presented in this paper generalize some known results. Examples are given for the illustration of the main results

Existence, multiplicity and classification results for solutions to kk-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)= \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 $k\in \mathbb N$. The function $r\rho'(r)/\rho(r)$ 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

On explosive solutions for a class of quasi-linear elliptic equations

We study existence, uniqueness, multiplicity and symmetry of large solutions for a class of quasi-linear elliptic equations. Furthermore, we characterize the boundary blow-up rate of solutions, including the case where the contribution of boundary curvature appears.Comment: 34 page