On critical exponents of a $k$-Hessian equation in the whole space

Yun Wang; Yutian Lei

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arXiv · 1602.06031

On critical exponents of a $k$-Hessian equation in the whole space

Yun Wang·Yutian Lei

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Abstract

In this paper, we study negative classical solutions and stable solutions of the following $k$-Hessian equation $$ F_k(D^2V)=(-V)^p \quad in~R^n $$ with radial structure, where $n \geq 3$, $1 1$. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research $k$-Hessian equations without radial structure.

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Yun Wang, Yutian Lei. 2016-02-19. On critical exponents of a $k$-Hessian equation in the whole space. https://arxiv.org/abs/1602.06031

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