An optimization problem for the first eigenvalue of the p-Laplacian plus a potential

Previous Article
CPAA Home
This Issue
Next Article
Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS

December 2006, 5(4): 675-690. doi: 10.3934/cpaa.2006.5.675

An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential

Julián Fernández Bonder ^1, and Leandro M. Del Pezzo ^2,

1.	Departamento de Matematica, FCEyN, UBA, 1428 Buenos Aires, Argentina
2.	Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina

Received January 2006 Revised May 2006 Published September 2006

Abstract
Related Papers

In this paper we study the optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential $V$ with respect to $V$, when the potential is restricted to a bounded, closed and convex set of $L^q(\Omega)$.

Keywords: minimization problems., nonlinear eigenvalues, $p-$Laplacian.

Mathematics Subject Classification: 35P15; Secondary: 49K2.

Citation: Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure & Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675

[1]	Leandro M. Del Pezzo, Julio D. Rossi. Eigenvalues for a nonlocal pseudo $p-$Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6737-6765. doi: 10.3934/dcds.2016093
[2]	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
[3]	Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219
[4]	Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431
[5]	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
[6]	Xianling Fan, Yuanzhang Zhao, Guifang Huang. Existence of solutions for the $p-$Laplacian with crossing nonlinearity. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1019-1024. doi: 10.3934/dcds.2002.8.1019
[7]	Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469
[8]	Ogabi Chokri. On the $L^p-$ theory of Anisotropic singular perturbations of elliptic problems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1157-1178. doi: 10.3934/cpaa.2016.15.1157
[9]	Guowei Dai, Ruyun Ma. Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 99-116. doi: 10.3934/dcds.2015.35.99
[10]	Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839
[11]	Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299
[12]	Raúl Ferreira, Julio D. Rossi. Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1469-1478. doi: 10.3934/dcds.2015.35.1469
[13]	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
[14]	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
[15]	Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020
[16]	Friedemann Brock, Leonelo Iturriaga, Justino Sánchez, Pedro Ubilla. Existence of positive solutions for $p$--Laplacian problems with weights. Communications on Pure & Applied Analysis, 2006, 5 (4) : 941-952. doi: 10.3934/cpaa.2006.5.941
[17]	Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075
[18]	C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545
[19]	Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173
[20]	Huijiang Zhao, Yinchuan Zhao. Convergence to strong nonlinear rarefaction waves for global smooth solutions of $p-$system with relaxation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1243-1262. doi: 10.3934/dcds.2003.9.1243

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences