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 and 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, 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, Indiana Univ. Math. J., (2004). doi: 10.1512/iumj.2004.53.2396. [3] P. Drábek and J. Milota, Methods of Nonlinear Analysis, Birkhäuser, 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, Calc. Var., (2007). doi: 10.1007/s00526-006-0055-8. [5] A. R. El Amrouss, Critical Point Theorems and Applications to Differential Equations, Acta Math. Sinica, English Series, (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 $p>2$, Adv.Differ Equ., (2002). [7] P. Takáč, On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue, 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, 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, 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, Indiana Univ. Math. J., (2004). doi: 10.1512/iumj.2004.53.2396. [3] P. Drábek and J. Milota, Methods of Nonlinear Analysis, Birkhäuser, 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, Calc. Var., (2007). doi: 10.1007/s00526-006-0055-8. [5] A. R. El Amrouss, Critical Point Theorems and Applications to Differential Equations, Acta Math. Sinica, English Series, (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 $p>2$, Adv.Differ Equ., (2002). [7] P. Takáč, On the Fredholm alternative for the $p$-Laplacian at the first eigenvalue, 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, J. Differ. Equ., (2002). doi: 10.1006/jdeq.2002.4173.
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