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Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem. (English) Zbl 0958.35054
Summary: 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 $T_1, \dots,T_k$ 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 $T_i$, $i=1,\dots,k$.

35J65Nonlinear boundary value problems for linear elliptic equations
35B25Singular perturbations (PDE)
58E05Abstract critical point theory