  This paper presents a new approach for addressing the singularly perturbed
nonlinear Schr\"odinger (NLS) equation:
  \begin{equation}
  -\varepsilon^2\Delta v + V(x) v =f(v),\ v>0,\ \lim_{|x|\to \infty} v(x)=0,
  \end{equation} where $V$ possesses a local maximum point and $f$ satisfies
the Berestycki-Lions conditions.The key to our approach is the derivation of a
refined lower bound on the gradient norm.
