Infinitely many sign-changing solutions of a critical fractional equation

TL;DR

This paper establishes the existence of infinitely many sign-changing solutions for certain critical fractional equations involving conformally invariant operators, using reduction techniques, blow-up analysis, and symmetry exploitation.

Contribution

It introduces a novel approach combining problem reduction, blow-up analysis, and symmetry methods to find multiple solutions for critical fractional equations.

Findings

Existence of an unbounded sequence of sign-changing solutions.

Nonexistence of positive solutions under certain conditions.

Development of new analytical techniques for fractional operators.

Abstract

In this paper, we obtain nonexistence results of positive solutions, and also the existence of an unbounded sequence of solutions that changing sign for some critical problems involving conformally invariant operators on the standard unit sphere, and the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together to blow up arguments, a variant of Pohozaev's type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere.