摘要
Let Ω R n be a bounded domain, H = L 2 （Ω）, L ： D（L） H → H be an unbounded linear operator, f ∈ C（ × R, R） and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf （x, u）, u ∈ D（L）, which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.
Let Ω R n be a bounded domain, H = L 2 （Ω）, L ： D（L） H → H be an unbounded linear operator, f ∈ C（ × R, R） and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf （x, u）, u ∈ D（L）, which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.
基金
Supported by the National Natural Science Foundation of Jiangsu Province of China (No. 10KJB110004 and 08KJD110010)
