New solutions for nonlinear Schrödinger equations with critical nonlinearity

We consider the following nonlinear Schrödinger equations in where V(r) is a radially symmetric positive function. In [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part I, Comm. Math. Phys. 235 (2003) 427–466], Ambrosetti, Malchiodi and Ni proved that if has a nondegenerate critical point r0≠0, then a layered solution concentrating near r0 exists. In this paper, we show that if and the dimension n=3,4 or 5, another new type of solution exists: this solution has a layer near r0 and a bubble at the origin.