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Multi-peak solutions for a wide class of singular perturbation problems. (English) Zbl 0922.35025

We are concerned with a wide class of singular perturbation problems arising from diverse fields such as phase transitions, chemotaxis, pattern formation, population dynamics and chemical reaction theory. We study the corresponding elliptic equations in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary has \(\overline{M}\) isolated, non-degenerate critical points. Then we show that for any positive integer \(m\leq\overline{M}\) there exists a stationary solution with \(M\) local peaks which are attained on the boundary and which lie close to these critical points. Our method is based on Lyapunov-Schmidt reduction.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
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