Liouville type theorem of integral equation with anisotropic structure

Abstract

In this paper, we classify all nonnegative solutions for the following integral equation:(0.1)u(x)=∫R+nKb(x,y)ynbf(u(y))dy, where b>1 is a constant. Here Kb(x,y) is the Green function of the following homogeneous Neumann boundary problem(0.2){−div(xnb∇u)=finR+n∂u∂xn=0on∂R+n. By using the method of moving planes in an integral form, we derive the symmetry of nonnegative solutions. We also establish the Liouville type theorem of (0.1). Similarly, the results can be generalized to the integral system.