Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions

We consider nonlinear second order elliptic problems of the type $-Δu=f(u) \text{ in } Ω, \qquad u=0 \text{ on } \partial Ω,$ where $Ω$ 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 $f(s)=λ(s-|s|^{p-1}s)$ with $p>1$ and $λ>λ_2(Ω)$ (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 $Ω$ 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 $Ω$ is either a ball or a square. In particular we give a complete description of the solution set for $λ\sim λ_2(Ω)$, computing the Morse index of the solutions.

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