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Uniqueness of multiple-spike solutions viat the method of moving planes. (English) Zbl 1141.35008

Summary: We study the uniqueness of multiple-spike solutions for some singularly perturbed Neumann problems in a ball. We completely classify all two-peaked solutions and, except in some degenerate situations, also all three-peaked solutions. Our main idea is using the method of moving planes to show that in the case of two peaks both of them must be located on a line containing the origin and for three peaks all of them must lie in a two-dimensional hyperplane containing the origin. Then we compute the degree of these solutions (restricted to a certain symmetry class) and show their uniqueness.

MSC:

35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35J40 Boundary value problems for higher-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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