-
Previous Article
On the pointwise jump condition at the free boundary in the 1-phase Stefan problem
- CPAA Home
- This Issue
-
Next Article
Method of the distance function to the Bence-Merriman-Osher algorithm for motion by mean curvature
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 |
$ -\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.
| [1] |
Marco Ghimenti, A. M. Micheletti. Non degeneracy for solutions of singularly perturbed nonlinear elliptic problems on symmetric Riemannian manifolds. Communications on Pure and 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 and 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 and Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237 |
| [4] |
Farid Bozorgnia, Martin Burger, Morteza Fotouhi. On a class of singularly perturbed elliptic systems with asymptotic phase segregation. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3539-3556. doi: 10.3934/dcds.2022023 |
| [5] |
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 |
| [6] |
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 |
| [7] |
Luigi Montoro. On the shape of the least-energy solutions to some singularly perturbed mixed problems. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1731-1752. doi: 10.3934/cpaa.2010.9.1731 |
| [8] |
Ahmed Bonfoh. Sufficient conditions for the continuity of inertial manifolds for singularly perturbed problems. Evolution Equations and Control Theory, 2022, 11 (4) : 1399-1454. doi: 10.3934/eect.2021049 |
| [9] |
Jaeyoung Byeon, Sang-hyuck Moon. Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1921-1965. doi: 10.3934/cpaa.2019089 |
| [10] |
Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535 |
| [11] |
Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1 |
| [12] |
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 and Continuous Dynamical Systems, 1996, 2 (2) : 221-236. doi: 10.3934/dcds.1996.2.221 |
| [13] |
Huiqing Zhu, Runchang Lin. $L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1493-1505. doi: 10.3934/dcdsb.2013.18.1493 |
| [14] |
Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431 |
| [15] |
Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 |
| [16] |
O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185 |
| [17] |
Dang Van Hieu, Le Dung Muu, Pham Kim Quy. New iterative regularization methods for solving split variational inclusion problems. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021185 |
| [18] |
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 |
| [19] |
Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041 |
| [20] |
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 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]