Multiplicity for critical and overcritical equations

TL;DR

This paper establishes existence and multiplicity of solutions for elliptic PDEs on compact manifolds by leveraging isometry invariances, addressing both critical and overcritical exponents.

Contribution

It introduces a novel approach using isometry invariances to obtain multiplicity results for elliptic PDEs with critical and overcritical exponents on manifolds.

Findings

Multiplicity results for equations with finite orbits and critical Sobolev exponent

Existence of multiple solutions for overcritical exponents without finite orbits

Application to Yamabe equation and similar elliptic PDEs

Abstract

On a Riemannian compact manifold, we give existence and multiplicity results for solutions of elliptic PDE by introducing isometry invariances. When the groups we used have finite orbits, we get multiplicity results for equations with the classical critical Sobolev exponent, for instance the Yamabe equation. When there is no finite orbits, the multiplicity is obtained for equations with overcritical exponents.