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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 Pohozaevidentities and blow-up analysis, we first prove a non-degeneracy result fordouble-tower solutions constructed in [18] in a suitable symmetric space. As anapplication, we obtain the existence of new type solutions for (0.1).

Non-degeneracy of double-tower solutions for nonlinear Schr\"odinger equation and applications