Even non-increasing solution for a Schrödinger type problem with Liouville–Weyl fractional derivative

Ledesma C.E.T.; Gutierrez H.C.; Rodríguez J.A.; Zhang Z. · Computational and Applied Mathematics (2022) · DOI: 10.1007/s40314-022-02124-6

Resumen

In this paper, we study the existence of even solution for a class of Schrödinger equations with Liouville–Weyl fractional derivatives {xD∞α(-∞Dxαu)=λg(u)inR,u∈I-α(R),where α∈(0,12), -∞Dxαu(·),xD∞αu(·) denote the left and right Liouville–Weyl fractional derivatives and g: R→ R is a continuous function satisfying suitable conditions. For the case that λ= 1 and g(u) = | u| p-1u- u with p∈(1,2α∗-1), we show that the above problem possesses at least one nonnegative and even solution. Besides, we also investigate the existence of normalized solutions for the first time to this problem when g is the form of more general nonlinearity.

Instituciones: Universidad Nacional de Trujillo