Picard iterates for positive solutions of a class of nonlinear eigenvalue problems
Positive solutions of the nonlinear eigenvalue problem \(F(u)=\lambda G(u)\) are studied with some nonlinear monotone operators F and G. In particular, the author considers the case of nonlinear elliptic differential equations of second order (nonlinear diffusion equations) \(F(u)=-div A(x,\text{grad} u)+b(x,u)=\lambda g(x,u)=\lambda G(u)\) with Dirichlet boundary conditions. Here, A(x,.) denotes a monotone vector field with respect to the second variable, while most authors restrict their considerations to the linear case with a positive definite Matrix A(x). Picard iterates are defined by \(u_ 0=0\) and \(F(u_{n+1})=\lambda G(u_ n).\) The author shows, that \(F(u)=\lambda G(u)\) has a positive solution iff \(\lambda\) belongs to some interval \((0,\lambda^*)\) or \((0,\lambda^*]\) with \(0<\lambda^*\leq +\infty\). In this case the Picard iterates converge from below to the minimal positive solution. The convergence is obtained by some comparison principles and the maximal monotonicity of the operators F and G, which replaces the usual Green's function arguments in the case of linear F. The endpoint \(\lambda^*\) is greater than the first eigenvalue of the problem linearized at the origin.
scientific article; zbMATH DE number 3859602