论文标题
dirichlet边界条件的均方根型问题的节点解决方案
Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions
论文作者
论文摘要
我们考虑类型的非线性二阶椭圆形问题\ [-ΔU= f（u）\ text {in}ω，\ qquad u = 0 \ qquad u = 0 \ text {on} \partialΩ，\ \]，其中$ω$是一个open $ c^{1,1} $ in $ c^{1,1} $ in $ \ mathbb in $ \ mathbb {r} $ n $ n $ c^n $，非线性包括sublrinear纯电源$ f（s）= | s |^{p-1} s $，$ 0 <p <1 $和allen-cahn type $ f（s）=λ（s- | s | s | s |^{p-1} s $，带有$ p> 1 $和$ p> 1 $ and $λ>λ>λ_2（lap）lape lap eig（lap）lich eig a lich eig lichele lichem eige licheen licheen。我们证明存在最少的能量节点（即更改符号）解决方案以及山间通道类型的淋巴结溶液。然后，我们给出明确的示例，其中相关级别不重合的域。对于$ω$是一个球或环，$ f $的情况是$ c^1 $，我们证明了级别的重合，而最少的能量淋巴结溶液是非放射线但轴向对称的功能。最后，如果$ω$是球或正方形，我们为Allen-Cahn型非线性提供了更强的结果。特别是，我们给出了$λ\simλ_2（ω）$的解决方案集的完整描述，计算解决方案的摩尔斯索引。
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.
