Rigorous Numerical Enclosures for Positive Solutions of Lane–Emden’s Equation with Sub-Square Exponents

The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane–Emden’s equation (Formula presented.) with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution u nearby a numerically computed approximation (Formula presented.) together with an explicit error bound, i.e., a bound for the difference between u and (Formula presented.) In particular, we focus on the sub-square case in which (Formula presented.) so that the derivative (Formula presented.) of the nonlinearity (Formula presented.) 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 (Formula presented.) on the unit square domain (Formula presented.).