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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 \(x\in T_j\), \(\min_{i\neq j}d(T_i,T_j) >2\max_{1\leq j\leq k}d(T_j, \partial\Omega)\), 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 \(T_i\), \(i=1,\dots,k\).

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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