论文标题
半线性椭圆方程的基态
Ground states of semilinear elliptic equations
论文作者
论文摘要
我们研究了$ΔU -f'（u）= 0 $的解决方案，其中潜在的$ f $可以在任意高度上有任意数量的井（包括具有亚临界衰变的无底井）。在我们的环境中，基态解决方案对应于能量最少的不稳定溶液。我们表明，在$ \ mathbb {r}^n $的凸域中，带有$ \ permatatorName {ric} \ geq 0 $的流形，地面状态始终是山间通用类型，并具有摩尔斯索引1。对于Allen -Cahn方程$ \ VAREPSILON^2ΔU -W'（u）= 0 $ on $ s^{n} $，我们证明地面状态是旋转的独特之处，并且与赤道相对应，作为最小的超曲面。我们还将基态能级分叉为$ \ varepsilon \ to 0 $，表明第一个$ n+1 $ min-min-max allen-cahn宽度为$ s^{n} $是基础状态，我们证明了对于相应的$（n+2）$ - th min-max max解决方案的间隙定理。
We study solutions of $Δu - F'(u)=0$, where the potential $F$ can have an arbitrary number of wells at arbitrary heights, including bottomless wells with subcritical decay. In our setting, ground state solutions correspond to unstable solutions of least energy. We show that in convex domains of $\mathbb{R}^N$ and manifolds with $\operatorname{Ric}\geq 0$, ground states are always of mountain-pass type and have Morse index 1. In addition, we prove symmetry of the ground states if the domain is either an Euclidean ball or the entire sphere $S^{N}$. For the Allen-Cahn equation $\varepsilon^2Δu - W'(u)=0$ on $S^{N}$, we prove the ground state is unique up to rotations and corresponds to the equator as a minimal hypersurface. We also study bifurcation at the energy level of the ground state as $\varepsilon\to 0$, showing that the first $N+1$ min-max Allen-Cahn widths of $S^{N}$ are ground states, and we prove a gap theorem for the corresponding $(N+2)$-th min-max solution.
