Entire Nodal Solutions with Prescribed Symmetry to Caffarelli-Kohn-Nirenberg Equations

TL;DR

This paper proves the existence of sign-changing solutions with prescribed symmetry for weighted critical p-Laplace equations of Caffarelli-Kohn-Nirenberg type, revealing multiple symmetry configurations and their incompatibilities.

Contribution

It introduces new methods to establish entire nodal solutions with various prescribed symmetries for weighted critical p-Laplace equations.

Findings

Existence of sign-changing solutions with prescribed symmetry

Multiple symmetry types for solutions

Conditions for symmetry incompatibility

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.