On some Sobolev and Pólya-Szegö type inequalities with weights and applications
Trung Hieu Giang, Nguyen Minh Tri, Dang Anh Tuan
TL;DR
The paper addresses a boundary-value problem for a degenerate elliptic operator $-\Delta_x u - |x|^{2\alpha} \partial^2_y u$ in a bounded 3D domain, and develops weighted Sobolev embeddings via a new Pólya-Szegö type inequality derived from a weighted isoperimetric inequality. It extends prior 2D results to 3D by establishing a weighted rearrangement inequality $\int_{\mathbb{R}^3} |\nabla_G u^*|^2 \le \int_{\mathbb{R}^3} |\nabla_G u|^2$, and proves a Sobolev-type weighted embedding with $\big(\int |x|^{2\alpha} |u|^6\big)^{1/6} \le C^{-1}_{2\alpha,6} (\int |\nabla_G u|^2)^{1/2}$, with a quantified lower bound on the best constant. Using these inequalities, the authors derive nonexistence results for a power-type nonlinearity via a Pohozaev identity on $G_\alpha$-star-shaped domains when $p>5$, and establish existence of nontrivial weak solutions under growth and regularity assumptions (A1)-(A5) through a Mountain Pass framework. The work advances the analysis of Grushin-type, degenerate elliptic problems in three dimensions and provides sharp, weight-aware embedding tools relevant to PDE existence theory.
Abstract
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), & \textrm{in } Ω, u = 0, & \textrm{on } \partial Ω, \end{cases} \end{align} where $x = (x_1, x_2) \in \mathbb{R}^2$, $Ω$ is a bounded smooth domain in $\mathbb{R}^3$, $(0,0,0) \in Ω$, and $α> 0$. 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.