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