Spectrum and constant sign solutions for a fractional Laplace problem with sign-changing weight

Abstract

In this paper, we study the spectrum of the fractional Laplace operator with sign-changing weight and show that there exist two simple, isolated principal eigenvalues λ1+ and λ1−. By use of the obtained spectrum results, we study the existence, multiplicity, and nonexistence of constant sign solutions to corresponding nonlinear problems according to the asymptotic behavior of nonlinear term f at 0, ∞, and whether f possesses zeros in R∖{0}. As far as we know, the spectral results presented here are new,and the rest theorems partially generalize the corresponding ones in the literature.