Existence and nonexistence of solutions for the fractional nonlinear Schrodinger equation.

ALARCÓN, SALOMÓN; RITORTO,ANTONELLA; ANALÍA SILVA

Mathematical Methods in the Applied Sciences.

Wiley

Año: 2021 vol. 44 p. 8903 - 8924

1099-1476

We consider the Fractional nonlinear Schrodinger equation with a potencial $V$. We study the existence andnonexistence of solutions concentrating at a local minimum point of Vwithout using any symmetry assumption on V . First, we prove thatcertain type of positive solutions exhibiting peaks do not exist. Then, westudy the existence of sign-changing solutions under a suitable conguration ofpositive and negative peaks. To guarantee the existence, we cannot neglect theinteraction between peaks. In particular, by using a minimization argument,we found solutions exhibiting peaks at the vertices of a 2l-regular polygon,such that two adjacent peaks have alternate sign.