Bubble solutions for an elliptic problem with critical growth in exterior domain

We consider the following nonlinear Schrödinger equation in R3R3{−Δu+λV(|y|)u=u5,u>0inΩεu=0,on∂Ωε where ε>0ε>0 is small, V(|y|)V(|y|) is smooth and 0λkλ>λk, where λkλk is the kkth eigenvalue of the Laplace–Beltrami operator in S3S3. We partially solve this question by constructing solutions with many bubbles, provided that λ≥a0k2λ≥a0k2 and ε>0ε>0 is small. Here a0>0a0>0 is a fixed constant.