New solutions for nonlinear Schrödinger equations with critical nonlinearity

We consider the following nonlinear Schrödinger equations in RnRn{ε2Δu−V(r)u+up=0in Rn;u>0in Rn and u∈H1(Rn), where V(r)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 M(r)=rn−1(V(r))p+1p−1−12 has a nondegenerate critical point r0≠0r0≠0, then a layered solution concentrating near r0r0 exists. In this paper, we show that if p=n+2n−2 and the dimension n=3,4n=3,4 or 5, another new type of solution exists: this solution has a layer near r0r0 and a bubble at the origin.