We make a formal study of the differential equation
$\begin{equation} u_{rr} + \frac{2}{r} u_r + \lambda u + u^{5+\varepsilon} = 0,\quad u_r(0) = u(1) = 0,\quad u > 0\quad \hbox{if}\ 0\mathrel{\hbox{{\char"36}}}r<1, \end{equation}$
when posed as a variational problem over a finite‐dimensional subset
$S_h$
of
$H^1_0$
comprising piecewise‐linear functions defined on a mesh of size
$h$
. We determine critical points
$U_h \in S_h$
of the variational form of (1). Such functions are perturbations of
$u$
when a solution of (1) exists, but we show that
$U_h$
can also exist when (1) has no solution and we determine an asymptotic expression for the solution branch
$(\lambda,U_h)$
when
$\Vert U_h\Vert _{\infty}$
is large and
$h \Vert U_h\Vert _{\infty}^2$
is small. If
$\varepsilon = 0$
, then
$u$
exists if
$\lambda > \mbox{{\char"19}}^2/4$
, and we give a formula expressing
$U_h$
as a perturbation of
$u$
. If
$\lambda \mathrel{\hbox{{\char"36}}}\mbox{{\char"19}}^2/4$
, then a solution of the differential equation does not exist, and
$U_h$
grows as
$h \rightarrow 0.$
We show that the rate of growth is proportional to
$h^{-1/4}$
if
$\lambda = \mbox{{\char"19}}^2/4$
, and
$h^{-1/3}$
if
$\lambda = 0$
. We compare these results with estimates for the solutions of (1) when
$\varepsilon \rightarrow 0^-$
. Our results are obtained by using formal asymptotic methods – particulary the method of matched asymptotic expansions – and are supported by some numerical calculations.
