Abstract
We consider the following singularly perturbed problem \begin{document}$ \begin{equation*} \varepsilon ^2 \Delta u - u + f(u) = 0, \, u>0 \text{ in } \Omega, \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial\Omega. \end{equation*} $\end{document} Existence of a solution with a spike layer near a min-max critical point of the mean curvature on the boundary \begin{document}$ \partial \Omega $\end{document} is well known when a nondegeneracy for a limiting problem holds. In this paper, we use a variational method for the construction of such a solution which does not depend on the nondengeneracy for the limiting problem. By a purely variational approach, we construct the solution for an optimal class of nonlinearities \begin{document}$ f $\end{document} satisfying the Berestycki-Lions conditions.
Highlights
Let Ω ⊂ Rn, n ≥ 3 be a bounded domain with a smooth boundary ∂Ω ∈ C4
∂ν which corresponds to steady states of a chemotaxis model of Keller and Segel [23] and the shadow system of Gierer and Meinhardt [18] for a pattern formation
In a classical paper [4], Berestycki and Lions proved the existence of radially symmetric least energy solution U of the limiting problem (2) when (f1),(f2) and the following (f3) are satisfied: (f3): there exists T > 0 satisfying F (T ) ≡
Summary
∂ν which corresponds to steady states of a chemotaxis model of Keller and Segel [23] and the shadow system of Gierer and Meinhardt [18] for a pattern formation. In a series of papers [28, 34, 35], Ni and his collaborators studied an asymptotic profile of the mountain pass solution as ε → 0 They proved very elegant results that for sufficiently small ε > 0, there exists a unique maximum point. (iii): for the mean curvature H of ∂Ω with respect to the outward unit normal vector field, lim H(xε) = min H(x) ε→0 x∈∂Ω when f satisfies the following additional conditions (f3-2): there exists μ > 2 such that μ t 0 f (s)ds. In a classical paper [4], Berestycki and Lions proved the existence of radially symmetric least energy solution U of the limiting problem (2) when (f1),(f2) and the following (f3) are satisfied:.
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