| The fractional Laplacian operator is a nonlocal operator,which has various applications in physics,probability,finance and others.Therefore,problems involving the fractional Laplacian have received considerable attention by many researchers.In this thesis,we are interested in the existence of positive solution for the following singular and nonlocal problem:where S2 ? RN is a bounded domain with smooth boundary ?Ω,N>2s,00,γ>0,a（x）∈ L1（Ω）,a（x）>0,f:[0,∞)→[0,∞)is an asymptotically linear continuous function,that isUsing variational methods,together with comparison principle,we prove that if the asymptotically linear term f satisfies some suitable monotonicity condition,there existsλ*=λ1,s/θ（here λ1,s is the first eigenvalue of（-Δ）s）,such that for each λ∈（0,λ*）,the problem has the least energy solution in both cases 0<γ<1 and γ≥1.Furthermore,we study the uniqueness and asymptotic behavior of the solution.In particular,we obtain the blow-up phenomenon of the solution when λ approaches λ*.At the same time,splitting in the cases 0<γ<1 and γ≥1,we also get the asymptotic behavior of the energy functional corresponding to the solution.Additionally,we show that the problem has no solution when λ≥λ*. |
