一个半线性特征值问题

A SEMILINEAR EIGENVALUE PROBLEM

考虑下面的二阶半线性特征值问题：这里Ｄ是ＲＮ中有界连通开区域，具有光滑边界，ｇ：Ｄ→Ｒ的有界连续函数，在Ｄ中达到正、负值．ｆ是上二阶连续可微的实函数，且满足ｋ１Ｘ≥ｆ（Ｘ）≥ｋ２Ｘ，ｋ１，ｋ２＞０，Ｘ＞０．我们得到在λ充分大时，（＊）有解；并且正数λ为（１）λ线性化方程的主特征值的充要条件是λ是（１）λ（２）的分歧点．（１）λ（２）有唯一分歧点（λ＞０时）．

In this paper, we discuss the following second semilinear eigenvalue problem:Here, D is a bounded connected open set in RN with smooth boundary. g is a continuous function from D into R a nd arrives at positive and negative values in D. fis a secondcontinuous differential function from R+ = [0,∞) into R and satisfy k1 x≥f(x)≥k2x, k1,k2 > 0, x∈R +.Equation (1)λ(2) is a semilinear eigenvalue problem with indefinite weight function,which satisfies Dirichlet boundary condition, while solving problem (* ) is to obtain itsregular solutions. We have known that the existence of solutions and the bifurcate of(1)λ (2) are closely connected with its linearization prblem as followingFor general situation, P. Hess and T. Kato have made some reseraches. here, we considerthe particular case (*) and obtain some further results.First,we point out that system (* * ) has only one positive principal eigenvalue λ*and give its describtion. Second, we prove that when λ> λ*, equation (*) has solutions.Finally, we give the following results: positive number λ is a principal eigenvalue of (* * ) ifand only if λ is a bifurcate point of equation (1)λ(2). Moreover, equation (1)λ(2) hasonly one bifurcate point (λ > 0). These results improve the results of P. aness and T. Kato.