Abstract
In this paper, we study the Kirchhoff elliptic equations of the form
$$
-M(\|\nabla_\lambda u\|^2)\Delta_\lambda u=w(x)f(u)
\quad \mbox{in }\mathbb R^{N},
$$
where $M$ is a smooth monotone function, $w$ is a weight function and $f(u)$
is of the form $u^p, e^u$ or $-u^{-p}$. The operator $\Delta_\lambda$ is strongly
degenerate and given by
$$
\Delta_\lambda=\sum_{j=1}^N \frac{\partial}{\partial x_j}\bigg(\lambda_j^2(x)\frac{\partial }{\partial x_j}\bigg).
$$
We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $\lambda_j$, $j=1,\ldots,N$.
Publisher
Uniwersytet Mikolaja Kopernika/Nicolaus Copernicus University
Subject
Applied Mathematics,Analysis