  We study the pure Neumann Lane-Emden problem in a bounded domain \[
  -\Delta u = |u|^{p-1} u \text{ in }\Omega, \qquad \partial_\nu u=0 \text{ on
}\partial \Omega, \] in the subcritical, critical, and supercritical regimes.
We show existence and convergence of least-energy (nodal) solutions (l.e.n.s.).
In particular, we prove that l.e.n.s. converge to a l.e.n.s. of a problem with
sign nonlinearity as $p\searrow 0$; to a l.e.n.s. of the critical problem as
$p\nearrow 2^*$ (in particular, pure Neumann problems exhibit no blowup
phenomena at the critical Sobolev exponent $2^*$); and we show that the limit
as $p\to 1$ depends on the domain. Our proofs rely on different variational
characterizations of solutions including a dual approach and a nonlinear
eigenvalue problem. Finally, we also provide a qualitative analysis of
l.e.n.s., including symmetry, symmetry-breaking, and monotonicity results for
radial solutions.
