  We consider the following nonlinear problem $$ (P) \quad \quad - \Delta u +
V(|y|)u=u^{p},\quad u>0 \quad \mbox{in} \ {\mathbb{R}}^N, \quad
  u \in H^1({\mathbb{R}}^N), $$ where $V(r)$ is a positive function, $1<p
<\frac{N+2}{N-2}$. We show that the multi-bump solutions constructed in [20] is
non-degenerate in a suitable symmetric space. We also use this non-degenerate
result to construct new solutions for (P).
