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

الأصل · EN

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