American Institute of Mathematical Sciences

Advanced
January  2010, 9(1): 233-248. doi: 10.3934/cpaa.2010.9.233

Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function

M. L. Miotto 1,

1. 

Departamento de Matemática, Universidade Federal de Santa Maria, 97105-900—Santa Maria RS, Brazil

Received  December 2008 Revised  May 2009 Published  October 2009

In this paper, existence and multiplicity results for semilinear elliptic equations in whole space, with concave and convex nonlinearities and weight function which can change sign, are established. The study is based on variational methods, more precisely, minimization techniques and mountain pass Theorem without Palais-Smale condition.
Keywords: critical Sobolev exponent., Multiple solutions, unbounded domains.
Mathematics Subject Classification: Primary: 35D05, 35B33; Secondary: 35A1.
Citation: M. L. Miotto. Multiple solutions for elliptic problem in $\mathbb{R}^N$ with critical Sobolev exponent and weight function. Communications on Pure and Applied Analysis, 2010, 9 (1) : 233-248. doi: 10.3934/cpaa.2010.9.233
[1]

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

Peng Chen, Xiaochun Liu. Multiplicity of solutions to Kirchhoff type equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2018, 17 (1) : 113-125. doi: 10.3934/cpaa.2018007
[3]

Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[4]

Yinbin Deng, Shuangjie Peng, Li Wang. Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 795-826. doi: 10.3934/dcds.2012.32.795
[5]

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

Mónica Clapp, Jorge Faya. Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3265-3289. doi: 10.3934/dcds.2019135
[7]

Salomón Alarcón. Multiple solutions for a critical nonhomogeneous elliptic problem in domains with small holes. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1269-1289. doi: 10.3934/cpaa.2009.8.1269
[8]

Wenmin Gong, Guangcun Lu. On Dirac equation with a potential and critical Sobolev exponent. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2231-2263. doi: 10.3934/cpaa.2015.14.2231
[9]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[10]

Y. Kabeya. Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition. Discrete and Continuous Dynamical Systems, 2006, 14 (1) : 117-134. doi: 10.3934/dcds.2006.14.117
[11]

Futoshi Takahashi. An eigenvalue problem related to blowing-up solutions for a semilinear elliptic equation with the critical Sobolev exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 907-922. doi: 10.3934/dcdss.2011.4.907
[12]

Antonio Capella. Solutions of a pure critical exponent problem involving the half-laplacian in annular-shaped domains. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1645-1662. doi: 10.3934/cpaa.2011.10.1645
[13]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[14]

F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure and Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355
[15]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[16]

T. Ogawa. The degenerate drift-diffusion system with the Sobolev critical exponent. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 875-886. doi: 10.3934/dcdss.2011.4.875
[17]

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

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[19]

Said Boulite, S. Hadd, L. Maniar. Critical spectrum and stability for population equations with diffusion in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 265-276. doi: 10.3934/dcdsb.2005.5.265
[20]

Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy