Eigenvalue Problem Of Fractional Laplacian Wit H Mixed Boundary Conditions And Its Application

In this thesis we first study the eigenvalue problem of fractional Laplacian with mixed boundary:where 0<s<1,? is a boundedC1,1 domain in Rd(d>2s)and B is homogeneous mixed boundary condition.Assuming that weight function a,b ? L?(?),and inf b(x)??>0,we obtain a series of eigenvalues that tend to be infinte and the x??principal eigenvalue.Then we analyze the continuous dependence of the principal eigenvalue ?1?[a,B]on the weight function a and the domain ?.Meanwhile,a prior estimate of principal eigenfunction u1?C?(?)for some ??(0,1)is obtained by Moser type iteration,we also get some inequalities associated with principal eigenvalue.Secondly,as an application of eigenvalue problem,the following Logistic type problem is considered Here h(x,u)satisfies some assumptions.We obtain that there exists a positive solution u* ? L?(?)of the nonlinear problem if and only if ?1?[a-?,B{<0 by constructing the pseudo-monotone operator and sup(sub)solution method.This results are nonlocal generalization of those of Cano-Casanova et.al.[JDE 2002],Fraile et.al.[JDE 1996 and Lopez-Gomez et.al.[JDE 1996]'s works.