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

Abstract

The present paper is concerned with negative classical solutions to a k-Hessian equation involving a nonlinearity with a general weight(P)$\{\begin{array}{cc}{S}_k\left(D^{2}u\right)=\lambda \rho \left(\left|x\right|\right){(1-u)}^q& \text{in }B,\\ u=0& \text{on }\partial B.\end{array}$ Here, B denotes the unit ball in $R^n$, $n>2k$, λ is a positive parameter and $q>k$ with $k\in N$. The function $r{\rho }^{\prime }\left(r\right)/\rho \left(r\right)$ satisfies very general conditions in the radial direction $r=\left|x\right|$. We show the existence, nonexistence, and multiplicity of solutions to Problem (P). The main technique used for the proofs is a phase-plane analysis related to a non-autonomous dynamical system associated to the equation in (P). Further, using the aforementioned non-autonomous system, we give a comprehensive characterization of $P_2$-, $P_3^{+}$-, $P_4^{+}$-solutions to the related problem($\hat{P}$)$\{\begin{array}{c}{S}_k\left(D^{2}w\right)=\rho \left(\left|x\right|\right){(-w)}^q,\\ w<0,\end{array}$ given on the entire space $R^n$. In particular, we describe new classes of solutions: fast decay $P_3^{+}$-solutions and $P_4^{+}$-solutions.