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August  2015, 35(8): 3771-3797. doi: 10.3934/dcds.2015.35.3771

## Concentrating solutions for an anisotropic elliptic problem with large exponent

 1 Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241 2 IECL, UMR 7502, Département de Mathématiques, Université de Lorraine, Bât. A, Ile de Saulcy, 57045 Metz Cedex 1, France

Received  September 2014 Revised  October 2014 Published  February 2015

We consider the following anisotropic boundary value problem $\nabla (a(x)\nabla u) + a(x)u^p = 0, \;\; u>0 \ \ \mbox{in} \ \Omega, \quad u = 0 \ \ \mbox{on} \ \partial\Omega,$ where $\Omega \subset \mathbb{R}^2$ is a bounded smooth domain, $p$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of anisotropic coefficient $a(x)$ on the existence of concentrating solutions. We show that at a given strict local maximum point of $a(x)$, there exist arbitrarily many concentrating solutions.
Citation: Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771
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##### References:
 [1] Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024 [2] Guanming Gai, Yuanyuan Nie, Chunpeng Wang. A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021070 [3] Namsu Ahn, Soochan Kim. Optimal and heuristic algorithms for the multi-objective vehicle routing problem with drones for military surveillance operations. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021037 [4] Haodong Chen, Hongchun Sun, Yiju Wang. A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2217-2242. doi: 10.3934/jimo.2020066 [5] José A. Carrillo, Bertram Düring, Lisa Maria Kreusser, Carola-Bibiane Schönlieb. Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3985-4012. doi: 10.3934/dcds.2021025 [6] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [7] Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038 [8] Annalisa Cesaroni, Valerio Pagliari. Convergence of nonlocal geometric flows to anisotropic mean curvature motion. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021065 [9] Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039 [10] Mirela Kohr, Sergey E. Mikhailov, Wolfgang L. Wendland. Dirichlet and transmission problems for anisotropic stokes and Navier-Stokes systems with L∞ tensor coefficient under relaxed ellipticity condition. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021042 [11] Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228 [12] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 [13] Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017 [14] Shuang-Lin Jing, Hai-Feng Huo, Hong Xiang. Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021113 [15] Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 [16] Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258 [17] Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195 [18] Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211 [19] Andreu Ferré Moragues. Properties of multicorrelation sequences and large returns under some ergodicity assumptions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2809-2828. doi: 10.3934/dcds.2020386 [20] Yuta Ishii, Kazuhiro Kurata. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021035

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