Nodal Solutions for Sublinear-Type Problems with Dirichlet Boundary Conditions

• Published in: International mathematics research notices 2022-02, Vol.2022 (5), p.3760-3804
• Main Authors: Bonheure, Denis, Moreira dos Santos, Ederson, Parini, Enea, Tavares, Hugo, Weth, Tobias
• Format: Article
• Language: English
• Subjects: Analysis of PDEs; Mathematics
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnaa233

Abstract We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$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 $0\lambda _2(\Omega )$ (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.