Eigenvalue Problem Of Fractional Laplacian Operator With Neumann Boundary Conditions And Applications

It is well known that the eigenvalues and eigenfunctions of linear differential op-erators are one of the core of operator theory and one of the foundations for studying corresponding non-linear problems.In this paper,the eigenvalue problem of weighted fractional Laplacian operators with Neumann boundary conditions （?） is studied,where λ>0,s ∈（0,1）,m,c ∈ C^0,1（Ω）.When c（x）= 0,the necessary and sufficient condition for the existence of positive principal eigenvalues for the above problems is that the weight function satisfies the con-dition ∫_Ωm（x）dx<0 and m（x₀）>0 for some x₀,and generalize the results of Montefusco et al.[Discrete Coitin.Dynam.Syst.B,2013]to s ∈（0,1）.We also prove the blow-up property of eigenvalues,that is,for a sequence of weighted functions {m_j}j=1∞ we prove the sufficient and necessary conditions for lim j>∞λ₁（m_j）=∞.This conclusion generalizes the classical Laplacian operator case[Umezu K.,et al.,Proc.R.Soc.Edinb.,2007]to the fractional Laplacian operator.When m（x）≡ 1,we prove the existence of principal eigenvalue λ₁ and its correspond-ing principal eigenvalue function.We also prove λ₁ is simple and any other solution of the above question with λ>λ₁,changes sign.Next,we study the logistic equation:It is proved that the necessary and sufficient condition for the existence of a unique positive solution is λ>λ₁ and the regularity of the positive solution.The results of Dirichlet boundary case[Alves M.O.,et aL.,Proc.R.Soc.Edinb.,2017]are extended to Neumann boundary case.