Solutions to general elliptic equations on nearly geodesically convex domains with many critical points

TL;DR

The authors address how the geometry of a domain on a Riemannian manifold influences the number of critical points of solutions to $-\ abla_g^2 u = f(q,u)$ with Dirichlet boundary conditions. They first construct Euclidean domains $\Omega_\varepsilon$ and torsion solutions $u_\varepsilon$ in which $u_\varepsilon$ has exactly $2n-1$ nondegenerate critical points (consisting of $n$ maxima and $n-1$ saddles) and show $\Omega_\varepsilon$ is almost convex. They then transplant this Euclidean configuration to a manifold via the exponential map, using a careful implicit-function-theorem argument to obtain a family $\widetilde{\Omega}_\eta$ with corresponding positive solutions $\tilde{u}_\eta$ that preserve the critical-point pattern. The notion of almost geodesic convexity is introduced to quantify how close these domains are to genuine geodesic convexity, and the authors prove the existence of sequences of domains around any point that are almost geodesically convex yet admit arbitrarily many critical points. Overall, the paper demonstrates the sharpness of convexity constraints in controlling critical points and provides a robust method to engineer prescribed nodal structures on manifolds through Euclidean models and perturbative analysis.

Abstract

Consider a complete $d$-dimensional Riemannian manifold $(\mathcal M,g)$, a point $p\in\mathcal M$ and a nonlinearity $f(q,u)$ with $f(p,0)>0$. We prove that for any odd integer $N\ge3$, there exists a sequence of smooth domains $Ω_k\subset\mathcal M$ containing $p$ and corresponding positive solutions $u_k:Ω_k\to\R^+$ to the Dirichlet boundary problem

Figures (1)

• Figure 1: The case $d=3$, $n=3$ and ${\varepsilon}=0.01$

Theorems & Definitions (19)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2
• proof
• Proposition 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof