Some Results of Hardy-Type Equation

In this article, we consider the Hardy-type equation as follows \(-Δ u=[φ(x)(1-|x|)]^α u^p, x∈ B_1(0)^N,\) where \(N≥ 3\), \(α\), \(p>0\) and \(φ(x)≥ 0\). We investigate the estimate, nonexistence, existence and uniqueness of positive solutions to this equation. Concretely, when \(1-2\), we give the estimate of positive solutions, and we obtain the existence of positive solutions of this equation with Dirichlet boundary condition. Moreover, we have some conclusions about nonexistence of positive solutions while \(p>1\) and \(α≤-2\). For the case \(0-2\), we establish the estimate of positive solutions as well. In addition, when \(0-2\).