Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains

In this paper, we are mainly concerned with the Dirichlet problems in exterior domains for the following elliptic equations: [Formula presented] with arbitrary r>0, where n≥2, 0<α≤2 and f(x,u) satisfies some assumptions. A typical case is the Hardy-Hénon type equations in exterior domains. We first derive the equivalence between (0.1) and the corresponding integral equations u(x)=∫Ω_rG_α(x,y)f(y,u(y))dy, where G_α(x,y) denotes the Green's function for [Formula presented] in Ω_r with Dirichlet boundary conditions. Then, we establish Liouville theorems for (0.2) via the method of scaling spheres developed in [17] by Dai and Qin, and hence obtain the Liouville theorems for (0.1). Liouville theorems for integral equations related to higher order Navier problems in Ω_r are also derived.