Schrödinger equations and coupled systems with Stein-Weiss convolution parts

Abstract

In this work, we investigate the existence of positive solutions for certain classes of Schrödinger equations and coupled systems with Stein-Weiss type nonlinearities. In the scalar case, we analyze classes of equations that involve perturbations in the Stein-Weiss term with a potential that may vanish at infinity or remain constant at 1. We consider both the case of a general nonlinearity with subcritical growth that satisfies certain appropriate conditions, and the critical homogeneous case in the sense of the Stein-Weiss inequality. Additionally, we explore two classes of coupled systems. The first class involves a linear system with potentials that may vanish at infinity and general nonlinearities with subcritical growth, also meeting specific conditions. The second class deals with a coupled nonlinear system, where the general nonlinearities exhibit critical exponential growth in the sense of the Trudinger-Moser inequality. We study the existence of positive solutions and the regularity of solutions for this system. To achieve these results, we employ variational methods, utilizing techniques such as minimization over the Nehari manifold, truncations combined with the penalization technique of Del Pino and Felmer, and Moser’s iteration method to obtain L∞−estimates. Furthermore, when dealing with the coupled nonlinear system, we present an alternative to the standard arguments, based on a variant of Palais symmetric criticality principle, instead of the traditional vanishing- nonvanishing and shifted sequences arguments of Lions, which are not applicable, due to the double weight present in the Stein-Weiss type convolution.