We will see the following singularly perturbed problem

A class of interesting solutions of this problem are solutions which concentrate and develop peaks around certain points in while vanishing elsewhere as . We call such solution as a spike layer solution. In recent decades, many papers have been published and solutions of various pattern (containing spike layer solutions) are constructed. Most of these results use Lyapunov-Schmidt reduction to construct solutions and rely on the uniqueness and nondegeneracy of the ground state solutions for

However, the nondegeneracy condition is difficult to prove in general nonlinearity case. We will prove the existence of a spike-layer solution concentrating at an isolated local minimum point of the potential via a variational method.\\

 

Reference:

(1) J. Byeon, L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Rational Mech. Anal. 2007.

(2) J. Byeon, L. Jeanjean, Erratum: "Standing waves for nonlinear Schödinger equations with a general nonlinearity'', Arch. Rational Mech. Anal. 2008.