Stable solutions and finite Morse index solutions of nonlinear elliptic equations with Hardy potential (Q393218)

The paper deals with Liouville-type results of stable solutions and finite Morse index solutions for the following nonlinear elliptic equation with Hardy potential NEWLINE\[NEWLINE \Delta u+{\mu\over{|x|^2}}u+|x|^l|u|^{p-1}u=0\quad\text{in}\;\Omega, NEWLINE\]NEWLINE where \(\Omega={\mathbb R}^N\) or \({\mathbb R}^N\setminus \{0\}\) with \(N\geq3,\) \(p>1,\) \(l>-2\) and \(\mu<(N-2)^2/4.\) The results obtained depend crucially on a new critical exponent \(p=p_c(l,\mu)\) and on the parameter \(\mu\) in the Hardy term. The authors prove that there exist no nontrivial stable solution and finite Morse index solution when \(1<p<p_c(l,\mu).\) It is shown also that for \(p> p_c (l,\mu)\) the equation admits a positive radial stable solution.