Symmetry and classification of positive solutions of some weighted elliptic equations

TL;DR

The paper studies radial symmetry and classification of positive solutions to the weighted elliptic equation $-\,\mathrm{div}(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u$ in $\mathbb{R}^N$, tied to Caffarelli-Kohn-Nirenberg inequalities. It establishes an exact equivalence between equations with $a$ on opposite sides of the critical value $a_c=\frac{N-2}{2}$, derives Liouville-type nonexistence results, obtains decay estimates via moving planes, and provides a complete radial classification in $\mathbf{L}^ fty_{Loc}$ with finite-energy extensions. Key contributions include a precise transform linking parameter sets when $a_1+a_2=2a_c$ and $b_2-b_1=b_1-a_1$, sharp nonexistence results at critical and near-critical $a$, universal decay rates of $|u(x)|\lesssim |x|^{-(N-2a-2)}$, and a full classification whereby nonzero solutions coincide with a bubble profile $u^*$ up to scaling (and translation in the isotropic case). These results extend classical symmetry and bubble-type classifications to the full weighted setting and clarify symmetry-breaking thresholds within the CKN framework.

Abstract

We study the weighted elliptic equation \begin{equation} -div(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u~~~\mbox{in}~\mathbb{R}^N ~~~~~~~~~~~~~~~~~~~~(0.1)\end{equation} with $N\geq 2$, which arises from the Caffarelli-Kohn-Nirenberg inequalities. Under the assumptions of finite energy and $a_1+a_2=N-2$, for nonnegative solutions we prove the equivalence between equation (0.1) with $a=a_1$ and equation (0.1) with $a=a_2$. Without finite energy assumptions, for $2\leq p<2^*$ we give the optimal parameter range in which nonnegative solutions of (0.1) in $\mathbf{L}^\infty_{Loc}(\mathbb{R}^N)$ must be radially symmetric, and give a complete classification for these solutions in this range.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof