Q-curvature flow with indefinite nonlinearity

TL;DR

This paper investigates the Q-curvature flow on the 4-sphere with indefinite nonlinearity, establishing conditions under which the prescribed Q-curvature problem admits solutions, involving critical point analysis and degree counting.

Contribution

It introduces new existence results for prescribed Q-curvature on S^4 with indefinite nonlinearity, utilizing critical point and degree theory methods.

Findings

Existence of solutions under specific critical point conditions.

Non-degeneracy of critical points influences solvability.

Additional degree counting condition ensures solution existence.

Abstract

In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $\Delta_{S^4} f\not=0$ at the saddle points and an extra condition such as a nontrivial degree counting condition.