Entire Nodal Solutions with Prescribed Symmetry to Caffarelli-Kohn-Nirenberg Equations
Edward Chernysh
TL;DR
The paper proves the existence of sign-changing, entire solutions to the weighted critical CK N equation $- abla\cdot\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)=|x|^{-bq}|u|^{q-2}u$ in $\mathbb{R}^n$, with $q=\frac{np}{n-p(1+a-b)}$, for $n\ge 4$, $1<p<n$, under $a<\frac{n-p}{p}$ and $a\le b<a+1$. It introduces a variational framework in $\mathcal{D}^{1,p}_a(\mathbb{R}^n,0)$ and constructs admissible symmetry groups $G_{\alpha,\mathfrak{m}}$ to obtain $(\alpha,\mathfrak{m})$-symmetric, sign-changing solutions for an infinite family of symmetry types, including the pinwheel-type symmetries encoded by $\alpha$ and $\mathfrak{m}$. The main contributions are (i) the first existence result for entire nodal solutions across the full parameter range, (ii) a classification mechanism distinguishing symmetry types via a binary coding and gcd arguments, and (iii) a symmetric Palais-Smale framework together with a symmetric Struwe-type decomposition yielding nontrivial solutions for the weighted problem, thereby extending known results for unweighted, Yamabe, and other related equations. The findings enable construction of infinitely many mutually incompatible symmetry configurations, enriching the landscape of nodal solutions for critical weighted elliptic problems with potential applications to geometric and PDE contexts.
Abstract
We establish the existence of sign-changing entire solutions to weighted critical $p$-Laplace equations of the Caffarelli-Kohn-Nirenberg type. In doing so, we investigate classes of symmetry and show that, for suitable symmetry configurations, there exists a non-trivial solution which changes sign and respects the corresponding prescribed symmetry. In addition, we describe conditions under which these symmetry-types are incompatible. Especially, we demonstrate the existence of entire nodal solutions for an infinite number of distinct symmetry types.