Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions

We consider nonlinear second order elliptic problems of the type $$-\mathrm{\Delta }u=f(u)\text{ in }\mathrm{\Omega },u=0\text{ on }\mathrm{\partial }\mathrm{\Omega },$$ where $\mathrm{\Omega }$ is an open $C^{1,1}$-domain in ${\mathbb{R}}^N$, $N\ge 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=\mathrm{\mid }s{\mathrm{\mid }}^{p-1}s$ with $0<p<1$ and of Allen-Cahn type $f(s)=\lambda (s-\mathrm{\mid }s{\mathrm{\mid }}^{p-1}s)$ with $p>1$ and $\lambda >{\lambda }_2(\mathrm{\Omega })$ (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 $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ 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.