Recently, for various elliptic problems of the type -εΔu = f(x), x ∈Ω; Bu = 0, x ∈ (partial deriv)Ω where Bu = u (Dirichlet boundary conditions) or Bu = (partial deriv)u/(partial deriv)v (Neumann boundary conditions), it has been proved that positive solutions with a single sharp peak or multiple sharp peaks exist when ε > 0 is sufficiently small. See, for example, [5, 12, 14] and the references therein. Since the space variable x does not appear in the nonlinearity, the topology and geometry of the domain Ω plays a central role in these problems.
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