Multiplicidad y concentración de soluciones para una ecuación de Schrödinger no-lineal unidimensional con frecuencia crítica: caso polinomial

Abstract

This work presents multiplicity and concentration results for a nonlinear one-dimensional Schrödinger equation with critical frequency, polynomial case:

ε2 v''(x) − V (x)v (x) + |v (x)| p−1 v (x) = 0 en x ∈ R, (P ε )

where V : R → R is non-negative continuous, explotes at infinity and Ω = {V = 0} = {0}. We focuse on the Polynomial Case as described by Byeon and Wang, where V decreases to polynomially to Ω. For α > 1 we find δα > 0 such that the limit problem, as ε → 0, is

u''(x) − P (x)u (x) + |u (x)|p−1 u (x) = 0 en x ∈ (−δα , δα ), (P y ) where P is a generalization of a homogeneous polynomial of even order and (−δα , δα ) is a neighborhood containing 0