Non-real Eigenvalues Of Linear Problems With Sign-changing Weight

In this paper,we study the non-real eigenvalues of linear eigenvalue problems with sign-changing weight.The main works are:In the first chapter,we consider the non-real eigenvalues of second-order indef-inite S-L eigenvalue problem with eigenparameter-dependent boundary conditionsτy:=-(p(x)y’)’ + q(x)y = λω(x)y,x ∈[0,1],B1y:= λy(0)cos θ1-p(0)y’(0)sin θ1 = 0,B2y:= λy(1)cos θ2-p(1)y’(1)sin θ2 = 0,where p,q,w is real-valued function and satisfy basic condition p(x)>0,ω(x)≠ 0 a.e.x ∈[0,1],1/p,q,ω ∈,L1[0,1],w(x)changes sign on[0,1],A is spectral parameter,and θ1,θ2 ∈[0,π).We obtain a priori bounds estimation for possible non-real eigenvalues and a sufficient condition for the existence and non-existence of non-real eigenvalues.In the second chapter,we consider the non-real eigenvalues of fourth-order in-definite p-Laplacian eigenvalues problemτy:=([y"]p-1)"+ q(x)[y]p-1=λω(x)[y]p-1 x ∈(0,1),y(0)= y(1)y"(0)= y"(1)=0,where p>1,[y]p-1= |y|p-2y,q and ω are real-valued functions and satisfy basic conditionω(x)≠ 0 a.e.x ∈[0,1],q,ω ∈ L2[0,1],w(x)changes sign on[0,1],A is spectral parameter.We obtain a priori bounds estimation for possible non-real eigenvalues and a sufficient condition for the non-existence of non-real eigenvalues.In the third chapter,we consider the non-real eigenvalues of fourth-order indef-inite elliptic eigenvalue problemτy:= △2y + ay = λwy,y ∈ L|ω|2(Ω),y|(?)Ω = △|(?)Ω= 0,where Ω(?)Rn(n ≥ 2)is bounded open set,with almost regular boundary and α,ω is real-valued function and satisfy basic conditionω(x)≠ 0 a.e.x:∈ Ω,ω ∈ L1,(Ω),a ∈ L∞(Ω),w(x)changes sign on Ω,λ is spectral parameter.We obtain a priori bounds for possible non-real eigenvalues and a sufficient condition for the existence and non-existence of non-real eigenvalues.