Classification of finite Morse index solutions to the polyharmonic Hénon equation (Q2093438)

The paper is devoted to the study of stable solutions and finite Morse index solutions (positive or sign-changing) of the following polyharmonic Hénon-type elliptic equation \[ -(\Delta)^m u=|u|^a|u|^{p-1}u,\quad\text{in }\mathbb{R}^n,\tag{1} \] where \(n\) is a large dimension, \(a\geq 0,\) integer \(m\geq 3\) and \(p>1\). The authors establish a non-existence result of homogeneous stable solutions to problem (1) for \(p\in \Big(\frac{n+2m+2a}{n-2m}, p_a(n, m)\Big)\), where \(p_a(n, m)\) is a critical exponent known as the generalized Joseph-Lundgren exponent. Then they outline proofs of a monotonicity formula and energy estimates. Finally they deal with the classification of the stable solutions to (1) and the finite Morse index solutions to (1), using blow-down analysis.