  This paper is concerned with the following nonlinear Schr\"odinger equation
\begin{equation} \label{eq} - \Delta u + V(|y|)u=u^{p},\quad u>0 \ \ \mbox{in}
\ \mathbb {R}^N, \ \ \ u \in H^1(\mathbb {R}^N), \end{equation} where $V(|y|)$
is a positive function, $1<p <\frac{N+2}{N-2}$. Based on the local Pohozaev
identities and blow-up analysis, we first prove a non-degeneracy result for
double-tower solutions constructed in [18] in a suitable symmetric space. As an
application, we obtain the existence of new type solutions for (0.1).
