American Institute of Mathematical Sciences

Advanced
2007, 2007(Special): 912-919. doi: 10.3934/proc.2007.2007.912

Mountain pass solutions to semilinear problems with critical nonlinearity

Ian Schindler 1, and  Kyril Tintarev 2,

1. 

Université des Sciences Sociales-UT1-Manufacture des Tabacs, 21 alles de Brienne, 31000 Toulouse, France
2. 

Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden

Received  September 2006 Revised  April 2007 Published  September 2007

The mountain pass statement for semilinear elliptic equations −$\Deltau = f(u)$ in $\mathbb{R}^N, N > 2$, with a critical exponent nonlinearity, namely $C_1 <= f(s)s/|s|^2^* <= C_2$, satisfies the (PS)$_c$ condition provided that the critical sequences are bounded and that the nonlinearity either has log-periodic oscillations or dominates its asymptotic values (relative to $|s|^2^*$ ) at zero and at infinity.
Keywords: critical exponent, Talenti solution, Semilinear elliptic equations, concentration compactness..
Mathematics Subject Classification: Primary 35J20, 35J6.
Citation: Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[1]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure &amp; Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[2]

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

Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure &amp; Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044
[4]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure &amp; Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103
[5]

Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure &amp; Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085
[6]

Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193
[7]

Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50
[8]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure &amp; Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[9]

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
[10]

Yanfang Peng. On elliptic systems with Sobolev critical exponent. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3357-3373. doi: 10.3934/dcds.2016.36.3357
[11]

Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351
[12]

Chao Ji, Vicenţiu D. Rădulescu. Concentration phenomena for magnetic Kirchhoff equations with critical growth. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021088
[13]

Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[14]

Jiayun Lin, Kenji Nishihara, Jian Zhai. Critical exponent for the semilinear wave equation with time-dependent damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4307-4320. doi: 10.3934/dcds.2012.32.4307
[15]

Mateus Balbino Guimarães, Rodrigo da Silva Rodrigues. Elliptic equations involving linear and superlinear terms and critical Caffarelli-Kohn-Nirenberg exponent with sign-changing weight functions. Communications on Pure &amp; Applied Analysis, 2013, 12 (6) : 2697-2713. doi: 10.3934/cpaa.2013.12.2697
[16]

Tsung-Fang Wu. On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function. Communications on Pure &amp; Applied Analysis, 2008, 7 (2) : 383-405. doi: 10.3934/cpaa.2008.7.383
[17]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[18]

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
[19]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure &amp; Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079
[20]

Lei Wei, Zhaosheng Feng. Isolated singularity for semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3239-3252. doi: 10.3934/dcds.2015.35.3239

 Impact Factor: 

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences