奇异椭圆型方程的特征值问题.pdf

The eigenvalue problem of a singular quasilinear elliptic equation Benjin Xuan ? Department of Mathematics University of Science and Technology of China Universidad Nacional de Colombia e-mail:wenyuanxbj@ Abstract In this paper, we shall study the eigenvalue problem of a singular quasi- linear elliptic equation. We will show that many results about the eigen- values and eigenfunctions in the non-singular case can be extended to the more general singular case, ., the ?rst eigenvalueλ 1is associated to a C 1,α(?) eigenfunction which is positive in ? and unique (up to a multi- plicative constant), that is,λ 1is simple. Moreoverλ 1is isolated, and is the unique positive eigenvalue associated to a non-negative eigenfunction. Fi- nally, we should prove some variational properties of the second eigenvalue λ 2. Key Words:singular quasilinear elliptic equation, eigenvalue problem, Ca?arelli-Kohn-Nirenberg inequality Mathematics Subject Classi?cations:35J60. 1 Introduction. In this paper, we shall study the eigenvalue problem of the following singular quasilinear elliptic equation: (?div (|x| ?ap|Du| p?2Du) =λ|x| ?(a+1)p+c|u| p?2u,in ? u= 0,on??, () ?Supported by Grant 10101024 and 10371116 from the National Natural Science Foundation of China. 1 _______________________________________________________________________________ 2 where ??R nis an open bounded domain withC 1boundary and 0∈?, 1< p < n,0≤a < n?p p , c >0. Fora= 0, c=p, there are many results about the eigenvalues and eigenfunc- tions of problem (), such asλ 1is associated to aC 1,α(?) eigenfunction which is positive in ? and unique (up to a multiplicative constant), that is,λ 1is sim- ple. Moreoverλ 1is isolated, and is the unique positive eigenvalue associated to a non-negative eigenfunction (cf. [LP, AT, CM] and references therein). In this paper, we will show that many results about the eigenvalues and eigen- functions in the case wherea= 0, c=pcan be extended to the more general case where 0≤a < n?