On Compactness Conditions for the p-Laplacian

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

On Compactness Conditions for the $p$-Laplacian

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

Received April 2014 Revised March 2015 Published February 2016

Abstract
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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).
[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.
[3]	P. Drábek and J. Milota, Methods of Nonlinear Analysis,, Birkh\, (2013). doi: 10.1007/978-3-0348-0387-8.
[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.
[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.
[6]	J. Fleckinger and P. Takáč, An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$,, \emph{Adv.Differ Equ.}, (2002).
[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.
[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.

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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).
[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.
[3]	P. Drábek and J. Milota, Methods of Nonlinear Analysis,, Birkh\, (2013). doi: 10.1007/978-3-0348-0387-8.
[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.
[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.
[6]	J. Fleckinger and P. Takáč, An improved Poincaré inequality and the $p$-Laplacian at resonance for $p>2$,, \emph{Adv.Differ Equ.}, (2002).
[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.
[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.

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