A study about elliptical partial differential equations of Choquard type

Abstract

In this project we present two elliptical Partial Differential Equation problems with a Choquard-type nonlinearity, which is characterized by the convolution between two functions, thus being a nonlocal nonlinearity, that is, it does not depend on the spatial variable. In our case, the first problem we present a Choquard equation related to the Schrodinger-Poisson system, in this case, together with the Choquard term we will have a weight function that changes sign and that is null in a subset of non-zero measure; we will first study an eigenvalue problem associated with this first problem and then use bifurcation results to obtain results of existence and non-existence of solutions. Regarding the second problem, we have a Choquard equation with the presence of the convection term, that is, a non-linearity that depends on the gradient; to guarantee the existence of a solution for this problem, we will study an associated evolution problem so that the solution of such problem converges (with respect to time) to a stationary function that is the solution of the original problem (AU)