Some results for semi-stable radial solutions of k-Hessian equations with weight on ℝ n

Abstract

We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u)=w(\left \vert x \right \vert )g(u)$ on the Euclidean space $\mathbb {R}^n$ . We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behaviour at infinity. All the estimates are given in terms of the spatial dimension $n$ , the values of $k$ and the behaviour at infinity of the growth rate function of $w$ .