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

Abstract

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 ) .