Existence and Error Estimates for Solutions of a Discrete Analog of Nonlinear Eigenvalue Problems

Finite-difference methods using the five-point discrete Laplacian and suitable boundary modifications for approximating <!-- MATH $(1) - \Delta u = \lambda f(x,u)$ --> in a plane domain on its boundary are considered. It is shown that if (1) has an isolated solution, , then the discrete problem has a solution, , for which <!-- MATH ${U_h} - u = O({h^2})$ --> . If the discrete problem has solutions, , such that <!-- MATH $|{U_h}| \leqq M$ --> as tends to zero, then (1) has a solution, , satisfying <!-- MATH $|u| \leqq M$ --> . Let <!-- MATH ${\lambda ^\ast}$ --> be a critical value of so that (1) has positive solutions for <!-- MATH $\lambda \leqq \lambda ^\ast$ --> but not for <!-- MATH $\lambda > {\lambda ^\ast}$ --> {\lambda ^\ast}$">, then the discrete problem has an analogous critical value <!-- MATH $\lambda _h^\ast$ --> and, under suitable conditions, <!-- MATH $\lambda _h^\ast - {\lambda ^\ast} = O({h^{4/3 - \epsilon}}),\epsilon > 0$ --> 0$">. Computed results for the case <!-- MATH $f(x,u) = {e^u}$ --> and the unit square are given.