$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data
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Well-posedness of low regularity solutions to the second order strictly hyperbolic equations with non-Lipschitzian coefficients
July 2019, 18(4): 1921-1965. doi: 10.3934/cpaa.2019089
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
Received August 2018 Revised August 2018 Published January 2019
Fund Project: The first author is supported by NRF-2017R1A2B4007816
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.
Keywords: Singular perturbation, Neumann condition, spike layer, mean curvature, transplantation flow, gradient flow, variational.
Mathematics Subject Classification: Primary: 35J20, 35J60, 35B25, 35B40.
Citation: Jaeyoung Byeon, Sang-hyuck Moon. Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1921-1965. doi: 10.3934/cpaa.2019089
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Jaeyoung Byeon Sang-hyuck Moon