Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential

We are concerned with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential: Δu+μ|x|2u+|x|l|u|p−1u=0inΩ, where Ω=RNΩ=RN, RN∖{0}RN∖{0} for N≥3N≥3, p>1p>1, l>−2l>−2 and μ<(N−2)2/4μ<(N−2)2/4. Our results depend crucially on a new critical exponent p=pc(l,μ)p=pc(l,μ) and the parameter μμ in the Hardy term. We prove that there exist no nontrivial stable solution and finite Morse index solution for 1