Multipeak solutions for asymptotically critical elliptic equations on Riemannian manifolds

Abstract

Let ( M , g ) be a smooth compact Riemannian manifold of dimension n ≥ 3 . We are concerned with the following asymptotically critical elliptic problem (0.1) Δ g u + a ( x ) u = u 2 ∗ − 1 − ε , u > 0 in M , where Δ g = − div g ( ∇ ) is the Laplace–Beltrami operator on M , a ( x ) is a C 1 function on M , 2 ∗ = 2 n n − 2 denotes the Sobolev critical exponent, ε is a small real parameter such that ε goes to 0. We use the Lyapunov–Schmidt reduction procedure to obtain that the problem (0.1) has a k -peaks solution for positive integer k ≥ 2 , which blow up and concentrate at some points in M .