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

This paper is devoted to studying the nonlinear problem with slightly subcritical and supercritical exponents ( S ± ε ) : Δ 2 u − c n Δ u + d n u = K u n + 4 n − 4 ± ε , u > 0 on S n , where n ≥ 5 , ε is a small positive parameter and K is a smooth positive function on S n . We construct some solutions of ( 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 + ε ) .