Ground states of semilinear elliptic problems with applications to the Allen–Cahn equation on the sphere

Abstract

We study solutions of $$\Delta u - F'(u)=0$$ Δ u - 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 $${\mathbb {R}}^N$$ R N and manifolds with $${\text {Ric}}\ge 0$$ Ric ≥ 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 $$S^{N}$$ S N . For the Allen–Cahn equation $$\varepsilon ^2\Delta u - W'(u)=0$$ ε 2 Δ u - W ′ ( u ) = 0 on $$S^{N}$$ 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 $$\varepsilon \rightarrow 0$$ ε → 0 , showing that the first $$N+1$$ N + 1 min-max Allen–Cahn widths of $$S^{N}$$ S N are ground states, and we prove a gap theorem for the corresponding $$(N+2)$$ ( N + 2 ) -th min-max solution.

The p-widths of a surface

“…Furthermore, he proved that 9 is some em-bedded minimal surface in S 3 having genus( 9 ) > 1 and area( 9 ) = ω 9 (S 3 , g S 3 ) ∈ (2π 2 , 8π). See also the related works [CGGM22,Hie20] computing low parameter phase-transition widths of the round three-sphere. However, to this point there had not been a single 4 (M, g) for which the areas ω p (M, g) (let alone the surfaces p ) are known for all p ∈ N * , not even in the two-dimensional case.…”