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We consider the following singularly perturbed problem ε2Δu − u + f(u) = 0, u > 0 in Ω, ∂u/∂ν = 0 on ∂Ω.Existence of a solution with a spike layer near a min-max critical point of the mean curvature on the boundary ∂Ω is well known when a nondegeneracy for a limiting problem holds. In this paper, we use a variational method for the construction of such a solution which does not depend on the nondengeneracy for the limiting problem. By a purely variational approach, we construct the solution for an optimal class of nonlinearities f satisfying the Berestycki-Lions conditions. [ABSTRACT FROM AUTHOR]

Abstract: Let Ω be a bounded domain in , , with the boundary . We consider the following singularly perturbed nonlinear elliptic problem on Ω where ν is an exterior normal to ∂Ω and a nonlinearity f of subcritical growth. Under rather strong conditions on f, it has been known that for small , there exists a solution of the above problem which exhibits a spike layer near local maximum points of the mean curvature H on ∂Ω as . In this paper, we obtain the same result under some conditions on f (Berestycki–Lions conditions), which we believe to be almost optimal. [Copyright &y& Elsevier]

Abstract: We consider the following singularly perturbed nonlinear elliptic problemwhere the nonlinearity f is of subcritical growth and is a bounded domain in . Under certain conditions of there exists a mountain pass solution for above problem and the solution exhibits a spike layer as . To see the location of the spike layer for small some additional assumptions, certain nondegeneracy for a limiting problem or monotonicity of have been assumed. In this paper, we characterize the location of the spike layer without such additional assumptions for . [Copyright &y& Elsevier]