American Institute of Mathematical Sciences

Advanced
May  2016, 15(3): 715-726. doi: 10.3934/cpaa.2016.15.715

On Compactness Conditions for the $p$-Laplacian

Pavel Jirásek 1,

1. 

Department of Mathematics, University of West Bohemia, Univerzitní 8, 306 14 Pilsen, Czech Republic

Received  April 2014 Revised  March 2015 Published  February 2016

We investigate the geometry and validity of various compactness conditions (e.g. Palais-Smale condition) for the energy functional \begin{eqnarray} J_{\lambda_1}(u)=\frac{1}{p}\int_\Omega |\nabla u|^p \ \mathrm{d}x- \frac{\lambda_1}{p}\int_\Omega|u|^p \ \mathrm{d}x - \int_\Omega fu \ \mathrm{d}x \nonumber \end{eqnarray} for $u \in W^{1,p}_0(\Omega)$, $1 < p < \infty$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$, $f \in L^\infty(\Omega)$ is a given function and $-\lambda_1<0$ is the first eigenvalue of the Dirichlet $p$-Laplacian $\Delta_p$ on $W_0^{1,p}(\Omega)$.
Keywords: variational methods, Fredholm alternative., Boundary value problems, Palais-Smale condition, $p$-Laplacian.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35P30, 46E3.
Citation: Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715
References:
[1]

J. Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids,, \emph{Comptes Rendus Acad.Sci. Paris Srie I}, (1987).   Google Scholar

[2]

P. Drábek, P. Girg, P. Takáč and M. Ulm, The Fredholm alternative for the $p$-Laplacian: bifurcation from infinity, existence and multiplicity,, \emph{Indiana Univ. Math. J.}, (2004).  doi: 10.1512/iumj.2004.53.2396.  Google Scholar

[3]

P. Drábek and J. Milota, Methods of Nonlinear Analysis,, Birkh\, (2013).  doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[4]

P. Drábek and P. Takáč, Poincaré inequality and Palais-Smale condition for the $p$-Laplacian,, \emph{Calc. Var.}, (2007).  doi: 10.1007/s00526-006-0055-8.  Google Scholar

[5]

A. R. El Amrouss, Critical Point Theorems and Applications to Differential Equations,, \emph{Acta Math. Sinica, (2005).  doi: 10.1007/s10114-004-0442-z.  Google Scholar

[6]

J. Fleckinger and P. Takáč, An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$,, \emph{Adv.Differ Equ.}, (2002).   Google Scholar

[7]

P. Takáč, On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue,, \emph{Indiana Univ. Math. J.}, (2002).  doi: 10.1512/iumj.2002.51.2156.  Google Scholar

[8]

P. Takáč, On the number and structure of solutions for a Fredholm alternative with the $p$-Laplacian,, \emph{J. Differ. Equ.}, (2002).  doi: 10.1006/jdeq.2002.4173.  Google Scholar

show all references

References:
[1]

J. Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids,, \emph{Comptes Rendus Acad.Sci. Paris Srie I}, (1987).   Google Scholar

[2]

P. Drábek, P. Girg, P. Takáč and M. Ulm, The Fredholm alternative for the $p$-Laplacian: bifurcation from infinity, existence and multiplicity,, \emph{Indiana Univ. Math. J.}, (2004).  doi: 10.1512/iumj.2004.53.2396.  Google Scholar

[3]

P. Drábek and J. Milota, Methods of Nonlinear Analysis,, Birkh\, (2013).  doi: 10.1007/978-3-0348-0387-8.  Google Scholar

[4]

P. Drábek and P. Takáč, Poincaré inequality and Palais-Smale condition for the $p$-Laplacian,, \emph{Calc. Var.}, (2007).  doi: 10.1007/s00526-006-0055-8.  Google Scholar

[5]

A. R. El Amrouss, Critical Point Theorems and Applications to Differential Equations,, \emph{Acta Math. Sinica, (2005).  doi: 10.1007/s10114-004-0442-z.  Google Scholar

[6]

J. Fleckinger and P. Takáč, An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$,, \emph{Adv.Differ Equ.}, (2002).   Google Scholar

[7]

P. Takáč, On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue,, \emph{Indiana Univ. Math. J.}, (2002).  doi: 10.1512/iumj.2002.51.2156.  Google Scholar

[8]

P. Takáč, On the number and structure of solutions for a Fredholm alternative with the $p$-Laplacian,, \emph{J. Differ. Equ.}, (2002).  doi: 10.1006/jdeq.2002.4173.  Google Scholar

[1]

Scott Nollet, Frederico Xavier. Global inversion via the Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 17-28. doi: 10.3934/dcds.2002.8.17
[2]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829
[3]

A. Azzollini. Erratum to: "On a functional satisfying a weak Palais-Smale condition". Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4987-4987. doi: 10.3934/dcds.2014.34.4987
[4]

Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729
[5]

John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515
[6]

Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143
[7]

Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623
[8]

Agnid Banerjee, Nicola Garofalo. On the Dirichlet boundary value problem for the normalized $p$-laplacian evolution. Communications on Pure & Applied Analysis, 2015, 14 (1) : 1-21. doi: 10.3934/cpaa.2015.14.1
[9]

Juan Campos, Rafael Obaya, Massimo Tarallo. Recurrent equations with sign and Fredholm alternative. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 959-977. doi: 10.3934/dcdss.2016036
[10]

R. Kannan, S. Seikkala. Existence of solutions to some Phi-Laplacian boundary value problems. Conference Publications, 2001, 2001 (Special) : 211-217. doi: 10.3934/proc.2001.2001.211
[11]

J. Ángel Cid, Pedro J. Torres. Solvability for some boundary value problems with $\phi$-Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 727-732. doi: 10.3934/dcds.2009.23.727
[12]

Nicola Abatangelo, Sven Jarohs, Alberto Saldaña. Positive powers of the Laplacian: From hypersingular integrals to boundary value problems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 899-922. doi: 10.3934/cpaa.2018045
[13]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020021
[14]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[15]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003
[16]

Anna Mercaldo, Julio D. Rossi, Sergio Segura de León, Cristina Trombetti. Behaviour of $p$--Laplacian problems with Neumann boundary conditions when $p$ goes to 1. Communications on Pure & Applied Analysis, 2013, 12 (1) : 253-267. doi: 10.3934/cpaa.2013.12.253
[17]

Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813
[18]

Alberto Cabada, J. Ángel Cid. Heteroclinic solutions for non-autonomous boundary value problems with singular $\Phi$-Laplacian operators. Conference Publications, 2009, 2009 (Special) : 118-122. doi: 10.3934/proc.2009.2009.118
[19]

Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252
[20]

Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055

2018 Impact Factor: 0.925

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences