June  2005, 4(2): 341-356. doi: 10.3934/cpaa.2005.4.341

A result on singularly perturbed elliptic problems

1. 

Departamento de Ingeniería Matemática, Universidad de La Frontera, Casilla 54-D, Temuco, Chile

2. 

Equipe de Mathématiques (UMR CNRS 6623), Université de Franche-Comté, 16 Route de Gray, 25030 Besançon, France

Received  June 2004 Revised  November 2004 Published  March 2005

We consider a class of equations of the form

$ -\varepsilon^2\Delta u + V(x)u = f(u), \quad u\in H^1(\mathbf R^N).$

For a local minimum $x_0$ of the potential $V(x)$, we show that there exists a sequence $\varepsilon_n\to 0$, for which corresponding solutions $u_n(x) \in H^1(\mathbf R^N) $ concentrate at $x_0$. Our assumptions on $f(\xi)$ are mainly the ones under which the associated autonomous problem

$ -\Delta v + V(x_0)v = f(v), \quad v\in H^1(\mathbf R^N),$

admits a non trivial solution.

Citation: Andrés Ávila, Louis Jeanjean. A result on singularly perturbed elliptic problems. Communications on Pure & Applied Analysis, 2005, 4 (2) : 341-356. doi: 10.3934/cpaa.2005.4.341
[1]

Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure & Applied Analysis, 2013, 12 (2) : 679-693. doi: 10.3934/cpaa.2013.12.679

[2]

Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823

[3]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237

[4]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. Dynamic boundary conditions as limit of singularly perturbed parabolic problems. Conference Publications, 2011, 2011 (Special) : 737-746. doi: 10.3934/proc.2011.2011.737

[5]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[6]

Luigi Montoro. On the shape of the least-energy solutions to some singularly perturbed mixed problems. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731

[7]

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

[8]

Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535

[9]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1

[10]

Daomin Cao, Norman E. Dancer, Ezzat S. Noussair, Shunsen Yan. On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 221-236. doi: 10.3934/dcds.1996.2.221

[11]

Huiqing Zhu, Runchang Lin. $L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1493-1505. doi: 10.3934/dcdsb.2013.18.1493

[12]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431

[13]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105

[14]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185

[15]

Nara Bobko, Jorge P. Zubelli. A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 1-21. doi: 10.3934/mbe.2015.12.1

[16]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499

[17]

Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041

[18]

Rong Xiao, Yuying Zhou. Multiple solutions for a class of semilinear elliptic variational inclusion problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 991-1002. doi: 10.3934/jimo.2011.7.991

[19]

Tomas Godoy, Alfredo Guerin. Existence of nonnegative solutions to singular elliptic problems, a variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1505-1525. doi: 10.3934/dcds.2018062

[20]

Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]