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

DOI: http://dx.doi.org/10.12775/TMNA.1999.020

#### 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=\hbox{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 $T_i$, $i=1,\ldots,k$.

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=\hbox{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 $T_i$, $i=1,\ldots,k$.

#### Keywords

Peaks; nonlinear; small diffusion; Conley index; Clarke derivative

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