American Institute of Mathematical Sciences

Advanced
2009, 2009(Special): 133-142. doi: 10.3934/proc.2009.2009.133

Some abstract critical point theorems and applications

Anna Maria Candela 1, and  Giuliana Palmieri 2,

1. 

Dipartimento Di Matematica, Universita' degli Studi di Bari "Aldo Moro", via E. Orabona 4, 70125 Bari, Italy
2. 

Dipartimento Di Matematica, Università degli Studi di Bari, Via E. Orabona 4, 70125 Bari, Italy

Received  July 2008 Revised  March 2009 Published  September 2009

Since Palais' pioneer paper in 1963, Condition $(C)$ in both the Palais--Smale version and Cerami's variant has been widely used in order to prove minimax existence theorems for $C^1$ functionals in Banach spaces. Here, we introduce a weaker version of these conditions so that a Deformation Lemma still holds and some critical points theorems can be stated. Such abstract results apply to $p$--Laplacian type elliptic problems.
Keywords: Critical point, $p$--Laplacian type equation, Deformation Lemma, symmetric functional, Mountain Pass Theorem, Linking Theorem, Condition $(C)$.
Mathematics Subject Classification: Primary: 58E05; Secondary: 35J35, 35J65.
Citation: Anna Maria Candela, Giuliana Palmieri. Some abstract critical point theorems and applications. Conference Publications, 2009, 2009 (Special) : 133-142. doi: 10.3934/proc.2009.2009.133
[1]

Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
[2]

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
[3]

Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045
[4]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[5]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[6]

Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345
[7]

Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683
[8]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
[9]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215
[10]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065
[11]

Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715
[12]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4593-4610. doi: 10.3934/dcds.2015.35.4593
[13]

E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209
[14]

V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413
[15]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775
[16]

M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703
[17]

Mohameden Ahmedou, Mohamed Ben Ayed, Marcello Lucia. On a resonant mean field type equation: A "critical point at Infinity" approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1789-1818. doi: 10.3934/dcds.2017075
[18]

Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063
[19]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[20]

Abdon E. Choque-Rivero, Iván Area. A Favard type theorem for Hurwitz polynomials. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 529-544. doi: 10.3934/dcdsb.2019252

 Impact Factor: 

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences