A subcritical approximation of the Paneitz problem on spheres

Abstract

This paper is concerned with the following subcritical approximation of a fourth order conformal invariant (the Paneitz curvature) on spheres \((S_\varepsilon ) : \Delta ^{2}u-c_n\Delta u+d_nu = K|u|^{\frac{8}{n-4}-\varepsilon }u\), in \( S^n\), where \(n\ge 5\), \( \varepsilon \) is a small positive parameter and K is a smooth positive function on \(S^n\). We construct some sign-changing solutions which blow up at two different critical points of K. Furthermore, we construct sign-changing solutions of \((S_\varepsilon )\) having two bubbles and blowing up at the same critical point of K.