On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand-Liouville equation

2020/08/17 by Fazly, Mostafa, Hu, Yeyao, Yang, Wen · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2008.07374

Abstract

We consider the nonlocal Hénon-Gelfand-Liouville problem (-Δ)^s u = |x|^a e^u\quadin \mathbb Rⁿ, for every s∈(0,1), a>0 and n>2s. We prove a monotonicity formula for solutions of the above equation using rescaling arguments. We apply this formula together with blow-down analysis arguments and technical integral estimates to establish non-existence of finite Morse index solutions when \dfracΓ(\frac n2)Γ(s)Γ((n-2s)/(2))(s+\frac a2)gt; \dfracΓ²((n+2s)/(4))Γ²((n-2s)/(4)).

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