Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy

Jaeyoung Byeon; Sang-hyuck Moon

doi:10.3934/cpaa.2019089

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2019, Volume 18, Issue 4: 1921-1965. Doi: 10.3934/cpaa.2019089

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Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy

Jaeyoung Byeon^, and
Sang-hyuck Moon

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

^* Corresponding author
^* Corresponding author

Received: July 31, 2018

Revised: July 31, 2018

Published: January 2019

The first author is supported by NRF-2017R1A2B4007816.

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Abstract

We consider the following singularly perturbed problem

$ \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*} $

Existence of a solution with a spike layer near a min-max critical point of the mean curvature on the boundary $ \partial \Omega $ 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.

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Mathematics Subject Classification: Primary: 35J20, 35J60, 35B25, 35B40.

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