Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem

Abstract

In this paper, we construct multipeak solutions for a singularly perturbed Dirichlet problem. Under the conditions that the distance function $d(x,\partial\Omega)$ has $k$ isolated compact connected critical sets $T_1,\ldots,T_k$ satisfying $d(x,\partial\Omega) =c_j=\text{const.}$, for all $x\in T_j$, $\min_{i\ne j}d(T_i,T_j)> 2\max_{1\le j\le k}d(T_j,\partial\Omega)$, and the critical group of each critical set $T_i$ is nontrivial, we construct a solution which has exactly one local maximum point in a small neighbourhood of $T_i$, $i=1,\ldots,k$.