Liouville theorems for the static Schrodinger equations with van der Waals type potentials

Abstract In this article, we are concerned with the static Schrödinger equation with the van der Waals potential. Such an equation can be transformed into a system of integral equations. We present the nonexistence of two classes of positive solutions. One is a class of locally bounded positive super-solutions, and the other is a class of positive solutions with some integrability. To prove the nonexistence of positive integrable solutions, we investigate their qualitative properties, involving the radial symmetry, the better integrability, the differentiability, and the decay estimates at infinity. Here, the regularity lifting lemma, the method of moving planes in integral forms, and the Pohozaev identity in integral forms play important roles.