American Institute of Mathematical Sciences

Advanced
July  2008, 7(4): 925-946. doi: 10.3934/cpaa.2008.7.925

Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions

Long Wei 1,

1. 

Department of Mathematics, East China Normal University, Shanghai 200062, China

Received  May 2007 Revised  November 2007 Published  April 2008

We investigate here the elliptic equation -div$(a(x)\nabla u)+a(x)u=0$ posed on a bounded smooth domain $\Omega$ in $\mathbb R^2$ with nonlinear Neumann boundary condition $\frac{\partial u}{\partial \nu}=\varepsilon e^u$, where $\varepsilon$ is a small parameter. We extend the work of Davila-del Pino-Musso [5] and show that if a family of solutions $u_\varepsilon$ for which $\varepsilon\int_{\partial Omega}e^{u_\varepsilon}$ is bounded, then it will develop up to subsequences a finite number of bubbles $\xi_i\in\partial Omega$, in the sense that $\varepsilon e^{u_\varepsilon}\to 2\pi\sum_{i=1}^k m_i\delta_{\xi_i}$ as $\varepsilon\rightarrow 0$ with $k, m_i \in \mathbb Z^+$. Location of blow-up points is characterized in terms of function $a(x)$.
Keywords: Exponential Neumann nonlinearity, concentration of solutions..
Mathematics Subject Classification: Primary: 35J65, 35J25; Secondary: 35J6.
Citation: Long Wei. Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 925-946. doi: 10.3934/cpaa.2008.7.925
[1]

Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301
[2]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[3]

Haitao Yang, Yibin Zhang. Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5467-5502. doi: 10.3934/dcds.2017238
[4]

Futoshi Takahashi. Singular extremal solutions to a Liouville-Gelfand type problem with exponential nonlinearity. Conference Publications, 2015, 2015 (special) : 1025-1033. doi: 10.3934/proc.2015.1025
[5]

Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198
[6]

Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431
[7]

Eugenia N. Petropoulou. On some difference equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2587-2594. doi: 10.3934/dcdsb.2017098
[8]

Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure & Applied Analysis, 2016, 15 (2) : 563-576. doi: 10.3934/cpaa.2016.15.563
[9]

Shuangjie Peng, Jing Zhou. Concentration of solutions for a Paneitz type problem. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1055-1072. doi: 10.3934/dcds.2010.26.1055
[10]

Manuel del Pino, Jean Dolbeault, Monica Musso. Multiple bubbling for the exponential nonlinearity in the slightly supercritical case. Communications on Pure & Applied Analysis, 2006, 5 (3) : 463-482. doi: 10.3934/cpaa.2006.5.463
[11]

A. Adam Azzam. Scattering for the two dimensional NLS with (full) exponential nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1071-1101. doi: 10.3934/cpaa.2018052
[12]

Jun Wang, Lu Xiao. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7137-7168. doi: 10.3934/dcds.2016111
[13]

Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
[14]

Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465
[15]

Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160
[16]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107
[17]

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044
[18]

Song Peng, Aliang Xia. Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1201-1217. doi: 10.3934/cpaa.2018058
[19]

Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685
[20]

Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences