Some mountain-pass solutions for small diffusion

Abstract

In this paper, we considerably improve results of Ni, Takagi, and Wei on the shape of the mountain-pass solution and the location of its maximum for certain boundary-value problems for certain weakly nonlinear elliptic equations with small diffusion, on domains in $\mathbb R^2$ or $\mathbb R^3$. We remove geometric conditions on the domain and nondegeneracy and uniqueness assumptions on the solutions.