(untitled) (Journal of Partial Differential Equations, 2024)
Let Ω be a smooth bounded domian in R^2 , H^1_0 (Ω) be the standard Sobolev space, and λ_f (Ω) be the first weighted eigenvalue of the Laplacian, namely, λ_f(Ω)=_uH^1_0(Ω),_Ωu^2dx=1_Ω| abla u|^2fdx,where f is a smooth positive function on Ω. In this paper, using blow-up analysis, we prove_uH^1_0(Ω),_Ω| abla u|^2fdx1_Ωe^4π fu^2(1+α||u||^2_2)dx