On some Sobolev and P\'olya-Szeg\"o type inequalities with weights and applications

We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations \begin{align}\tag{P}\label{P} \begin{cases} - \Delta_x u - |x|^{2\alpha} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u) & \textrm{in } \Omega, u = 0 & \textrm{on } \partial \Omega, \end{cases} \end{align} where $x = (x_1, x_2) \in \mathbb{R}^2$, $\Omega$ is a bounded smooth domain in $\mathbb{R}^3$, $(0,0,0) \in \Omega$, and $\alpha > 0$. In this paper, we will study this problem by establishing embedding theorems for weighted Sobolev spaces. To this end, we need a new P\'olya-Szeg\"o type inequality, which can be obtained by studying an isoperimetric problem for the corresponding weighted area. Our results then extend the existing ones in \cite{nga, Luyen2} to the three-dimensional context.