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

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 $N\ge 3$, there exists a sequence of smooth domains $\hat{I}{\text{c◯}}_k\subset \mathcal{M}$ containing $p$ and corresponding positive solutions $u_k:\hat{I}{\text{c◯}}_k\to {\mathbb{R}}^{+}$ to the Dirichlet boundary problem