American Institute of Mathematical Sciences

Advanced
July  2009, 8(4): 1269-1289. doi: 10.3934/cpaa.2009.8.1269

Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes

Salomón Alarcón 1,

1. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

Received  April 2008 Revised  January 2009 Published  March 2009

We consider the problem: $-\Delta u=|u|^{\frac{4}{N-2}}u+\varepsilon f(x)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega\subset R^N$ is a bounded smooth domain which exhibits small holes, $f\geq 0$, $f$ is not equivalent to $0$ and $\varepsilon>0$ is small. Using the reduction method and a min-max scheme worked out with topological arguments, we construct multiple solutions by gluing negative double-spike patterns located near each of the holes.
Keywords: solution with multiple double-spikes, reduction method, Nonhomogeneous elliptic equation, critical Sobolev exponent, small holes..
Mathematics Subject Classification: Primary: 35B40, 35B45; Secondary: 35J60, 47J3.
Citation: Salomón Alarcón. Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1269-1289. doi: 10.3934/cpaa.2009.8.1269
[1]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[2]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
[3]

M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure & Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233
[4]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[5]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[6]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[7]

Guangze Gu, Xianhua Tang, Youpei Zhang. Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3181-3200. doi: 10.3934/cpaa.2019143
[8]

A. M. Micheletti, Monica Musso, A. Pistoia. Super-position of spikes for a slightly super-critical elliptic equation in $R^N$. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 747-760. doi: 10.3934/dcds.2005.12.747
[9]

Leonid Berlyand, Volodymyr Rybalko. Homogenized description of multiple Ginzburg-Landau vortices pinned by small holes. Networks & Heterogeneous Media, 2013, 8 (1) : 115-130. doi: 10.3934/nhm.2013.8.115
[10]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[11]

Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698
[12]

Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603
[13]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[14]

Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[15]

M. M. Cavalcanti, V.N. Domingos Cavalcanti, D. Andrade, T. F. Ma. Homogenization for a nonlinear wave equation in domains with holes of small capacity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 721-743. doi: 10.3934/dcds.2006.16.721
[16]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117
[17]

Michinori Ishiwata. Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent. Conference Publications, 2005, 2005 (Special) : 443-452. doi: 10.3934/proc.2005.2005.443
[18]

M. Ben Ayed, Abdelbaki Selmi. Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1705-1722. doi: 10.3934/cpaa.2010.9.1705
[19]

Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217
[20]

José Carmona, Pedro J. Martínez-Aparicio. Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 15-31. doi: 10.3934/dcds.2017002

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences