We are concerned with the normalized \(\ell \) -peak solutions to the nonlinear Schrödinger equation \(\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta v+V(x)v=f(v)+\lambda v,\\ \int _{\mathbb {R}^N}v^2 =\alpha \varepsilon ^N. \end{array}\right. } \end{aligned}\) Here \(\lambda \in \mathbb {R}\) will arise as a Lagrange multiplier, V has a local maximum point, and f is a general \(L^2\) -subcritical nonlinearity that satisfies a nonlipschitzian property such that \(\lim _{s\rightarrow 0} f(s)/s=-\infty \) . The peaks of solutions that we construct cluster around a local maximum of V as \(\varepsilon \rightarrow 0\) . Since there is no information about the uniqueness or nondegeneracy of the limiting system, a sensitive lower gradient estimate should be made when the local centroids of the functions are away from the local maximum of V. We introduce a new method to obtain this estimate, which differs significantly from the ideas of del Pino and Felmer [22] (Math. Ann. 2002), where a special gradient flow with high regularity is used, and in Byeon and Tanaka [7, 8] (J. Eur. Math. Soc. 2013 & Mem. Amer. Math. Soc. 2014), where an additional translation flow is introduced. We also give the existence of ground state solutions for the autonomous problem, i.e., the case \(V\equiv 0\) . The ground state energy is not always negative and the strict subadditivity of the ground state energy is achieved here by strict concavity.
