Blow-up solutions for Paneitz–Branson type equations with critical growth

Abstract

Let (M,g) be a smooth, compact Riemannian manifold of dimension n≥7. We consider the Paneitz–Branson type equation

Δ_g²u−div_g(A du)+au=|u|^{2♯−2−ε}u in M,

where Δ_g=−div_g∇ is the Laplace–Beltrami operator, A is a smooth symmetrical (2,0)-tensor fields, a is a smooth function on M, 2^♯=2n/(n−4) is the critical exponent for the Sobolev embedding and ε is a small positive parameter. Under suitable conditions on the Tr_gA, we construct solutions u_ε which blow up at one point of the manifold as ε goes to zero.