On some Sobolev and Pólya-Szegö 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} - Δ_x u - |x|^{2α} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u) & $\text{in }$ Ω, u = 0 & $\text{on }$ \partial Ω, \end{cases} \end{align} where $x=(x_1,x_2)\in {\mathbb{R}}^2$, $\mathrm{\Omega }$ is a bounded smooth domain in ${\mathbb{R}}^3$, $(0,0,0)\in \mathrm{\Omega }$, 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 Pólya-Szegö 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.