A New Approach to Solving Singularly Perturbed NLS at Local Potential Maxima

Abstract

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.