  The purpose of this paper is to obtain rigorous numerical enclosures for
solutions of Lane-Emden's equation $-\Delta u=|u|^{p-1} u$ with homogeneous
Dirichlet boundary conditions. We prove the existence of a nondegenerate
solution $u$ nearby a numerically computed approximation $\hat{u}$ together
with an explicit error bound, i.e., a bound for the difference between $ u $
and $\hat{u}$. 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 $|u|^{p-1} u$ 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
$\Omega=(0,1)^2$.
