Existence and nonexistence of solutions for an approximation of the Paneitz problem on spheres

Abstract

This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents $(S_{\pm \varepsilon}): \Delta ^{2}u-c_{n}\Delta u+d_{n}u = Ku^{ \frac{n+4}{n-4}\pm \varepsilon}$ ( S ± ε ) : Δ 2 u − c n Δ u + d n u = K u n + 4 n − 4 ± ε , $u>0$ u > 0 on $S^{n}$ S n , where $n\geq 5$ n ≥ 5 , ε is a small positive parameter and K is a smooth positive function on $S^{n}$ S n . We construct some solutions of $(S_{-\varepsilon})$ ( S − ε ) that blow up at one critical point of K. However, we prove also a nonexistence result of single-peaked solutions for the supercritical equation $(S_{+\varepsilon})$ ( S + ε ) .