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 $0 1$ 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.

中文翻译：

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具有狄利克雷边界条件的亚线性类型问题的节点解

我们考虑类型为 \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] 的非线性二阶椭圆问题，其中 $\Omega $ 是 $\mathbb{R}^N$, $N\geq 2$ 中的开放 $C^{1,1}$ 域，在非线性的一些一般假设下，包括亚线性纯幂的情况 $ f(s)=|s|^{p-1}s$ 与 $0 1$ 和 $\lambda>\lambda_2(\Omega)$（拉普拉斯算子的第二个狄利克雷特征值）。我们证明了最小能量节点（即符号变化）解和山口型节点解的存在。然后，我们给出了相关级别不一致的域的明确示例。对于 $\Omega$ 是球或环且 $f$ 属于 $C^1$ 类的情况，我们证明这些能级是重合的，并且最小能量节点解是非径向但轴对称的函数。最后，在 $\Omega$ 是球或正方形的情况下，我们为 Allen-Cahn 型非线性提供了更强的结果。特别地，我们给出了 $\lambda\sim \lambda_2(\Omega)$ 的解决方案集的完整描述，计算了解决方案的莫尔斯指数。