On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem

Imene Bendahou; Zied Khemiri; Fethi Mahmoudi

doi:10.3934/dcds.2020118

Article Contents

2020, Volume 40, Issue 4: 2367-2391. Doi: 10.3934/dcds.2020118

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On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem

Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Campus Universitaire, 2092 Tunis El Manar, Tunisia

^* Corresponding author: F. Mahmoudi
^* Corresponding author: F. Mahmoudi

Received: July 2019

Revised: November 2019

Published: January 2020

F. Mahmoudi is supported by Fondecyt Grant 1180526, CONICYT.

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Abstract

Given a smooth bounded domain $ \Omega \subset \mathbb {R}^n $ and consider the problem

$ \left\{\begin{array} {cccccc} - \Delta u = |u|^p - \sigma &\hbox{in } \Omega \\ \dfrac{\partial u}{\partial \nu} = 0 &\hbox{on}\ \partial \Omega \end{array}\right. $

where $ p $ is subcritical exponent ($ p > 1 $ if $ n = 2 $ and $ 1 < p < \frac{n+2}{n-2} $ if $ n \geq 3 $), $ \sigma > 0 $ is a large parameter and $ \nu $ denotes the outward normal of $ \partial\Omega $. Let $ \Gamma $ be an interior straighline intersecting orthogonally with $ \partial\Omega $. Assuming moreover that $ \Gamma $ satisfies a non-degeneracy condition, we construct a new class of solutions which consist of large number of spikes concentrating on $ \Gamma $, showing as in [5,6] that higher dimensional concentration can exist without resonance condition.

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Mathematics Subject Classification: Primary: 35J20, 35J25; Secondary: 35J60, 35J61.

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