一类凹凸型函数的半线性椭圆型方程的非平凡解的个数问题

The Multiplicities of the Nontrivial Solutions of a Class of Semilinear Elliptic Equations with the Type of Concave- convex Functions

研究了一类凹凸型函数的半线性椭圆型方程:-Δu=λu+g(u),x∈Ω;u=0,x∈Ω的非平凡解的个数问题,利用函数g关于变量u的凹凸性质、椭圆型方程的Dirichlet边值的本征值问题的比较原理以及Leray-Schauder度关于孤立解指标的计算,得到了该类方程的非平凡解的个数.

In this paper, the multiplicities of the nontrivial solutions of a type of semilinear elliptic equations with functions like concave - convex ones on the unknown solutions are studied. By using the property of the functiong g and the comparison principles about the eigenvalues of the elliptic equations with Dirichlet boundary conditions, the precise index of the Leray - Schauder degree of the nontrivial solutions are computed, which determines the concrete multiplicities of the trivial or nontrivial solutions.