# American Institute of Mathematical Sciences

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

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 $W_0^{1,p}(\Omega)$.
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:
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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). 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
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