  We consider nonlinear second order elliptic problems of the type \[ -\Delta
u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where
$\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some
general assumptions on the nonlinearity that include the case of a sublinear
pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen-Cahn type
$f(s)=\lambda(s-|s|^{p-1}s)$ with $p>1$ and $\lambda>\lambda_2(\Omega)$ (the
second Dirichlet eigenvalue of the Laplacian). We prove the existence of a
least energy nodal (i.e. sign changing) solution, and of a nodal solution of
mountain-pass type. We then give explicit examples of domains where the
associated levels do not coincide. For the case where $\Omega$ is a ball or
annulus and $f$ is of class $C^1$, we prove instead that the levels coincide,
and that least energy nodal solutions are nonradial but axially symmetric
functions. Finally, we provide stronger results for the Allen-Cahn type
nonlinearities in case $\Omega$ is either a ball or a square. In particular we
give a complete description of the solution set for $\lambda\sim
\lambda_2(\Omega)$, computing the Morse index of the solutions.
