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Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations
Boundary Value Problems volume 2006, Article number: 41295 (2006)
Abstract
The goal of this paper is to study the existence and the multiplicity of non-trivial weak solutions for some degenerate nonlinear elliptic equations on the whole space. The solutions will be obtained in a subspace of the Sobolev space. The proofs rely essentially on the Mountain Pass theorem and on Ekeland's Variational principle.
References
Alves CO, Gonçalves JV, Miyagaki OH:On elliptic equations in with critical exponents. Electronic Journal of Differential Equations 1996,1996(9):1-11.
Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. Journal of Functional Analysis 1973,14(4):349-381. 10.1016/0022-1236(73)90051-7
Brezis H: Analyse Fonctionnelle. Théorie et Applications, Collection of Applied Mathematics for the Master's Degree. Masson, Paris; 1983.
De Nápoli P, Mariani MC:Mountain pass solutions to equations of-Laplacian type. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2003,54(7):1205-1219.
Díaz JI: Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equations, Research Notes in Mathematics. Volume 106. Pitman, Massachusetts; 1985.
do Ó JMB: Existence of solutions for quasilinear elliptic equations. Journal of Mathematical Analysis and Applications 1997,207(1):104-126. 10.1006/jmaa.1997.5270
do Ó JMB:Solutions to perturbed eigenvalue problems of the-Laplacian in. Electronic Journal of Differential Equations 1997,1997(11):1-15.
Ekeland I: On the variational principle. Journal of Mathematical Analysis and Applications 1974,47(2):324-353. 10.1016/0022-247X(74)90025-0
Gonçalves JV, Miyagaki OH:Multiple positive solutions for semilinear elliptic equations in involving subcritical exponents. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1998,32(1):41-51.
Mihăilescu M, Rădulescu V: Ground state solutions of non-linear singular Schrödinger equations with lack of compactness. Mathematical Methods in the Applied Sciences 2003,26(11):897-906. 10.1002/mma.403
Motreanu D, Rădulescu V: Eigenvalue problems for degenerate nonlinear elliptic equations in anisotropic media. Boundary Value Problems 2005,2005(2):107-127. 10.1155/BVP.2005.107
Pflüger K:Existence and multiplicity of solutions to a-Laplacian equation with nonlinear boundary condition. Electronic Journal of Differential Equations 1998,1998(10):1-13.
Rabinowitz PH: On a class of nonlinear Schrödinger equations. Zeitschrift für Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathématiques et de Physique Appliquées 1992,43(2):270-291.
Rădulescu V, Smets D: Critical singular problems on infinite cones. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2003,54(6):1153-1164.
Şt. Cîrstea F, Rădulescu V: Multiple solutions of degenerate perturbed elliptic problems involving a subcritical Sobolev exponent. Topological Methods in Nonlinear Analysis 2000,15(2):283-300.
Tarantello G: On nonhomogeneous elliptic equations involving critical Sobolev exponent. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1992,9(3):281-304.
Willem M: Analyse harmonique réelle, Methods Collection. Hermann, Paris; 1995.
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Mihăilescu, M. Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations. Bound Value Probl 2006, 41295 (2006). https://doi.org/10.1155/BVP/2006/41295
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DOI: https://doi.org/10.1155/BVP/2006/41295