LAUSR

The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane– Emden’s equation −∆u = |u|p−1u with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution u nearby a numerically computed approximation û together with an explicit error bound, i.e., a bound for the difference between u and û. In particular, we focus on the sub-square case in which 1 < p < 2 so that the derivative p|u|p−1 of the nonlinearity |u|p−1u is not Lipschitz continuous. In this case, it is problematic to apply the classical Newton-Kantorovich theorem for obtaining the existence proof, and moreover several difficulties arise in the procedures to obtain numerical integrations rigorously. We design a method for enclosing the required integrations explicitly, proving the existence of a desired solution based on a generalized Newton-Kantorovich theorem. A numerical example is presented where an explicit solution-enclosure is obtained for p = 3/2 on the unit square domain Ω = (0, 1).