EXISTENCE AND MULTIPLICITY OF WEAK SOLUTIONS FOR A CLASS OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS ˘ MIHAI MIHAILESCU Received 11 January 2005; Revised 4 July 2005; Accepted 17 July 2005 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 RN . The solutions will be obtained in a subspace of the Sobolev space W 1,p (RN ). The proofs rely essentially on the Mountain Pass theorem and on Ekeland’s Variational principle. Copyright © 2006 Mihai Mih˘ ilescu. This is an open access article distributed under. | EXISTENCE AND MULTIPLICITY OF WEAK SOLUTIONS FOR A CLASS OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS MIHAI MIHAILESCU Received 11 January 2005 Revised 4 July 2005 Accepted 17 July 2005 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 RN. The solutions will be obtained in a subspace of the Sobolev space IV1 p RN . The proofs rely essentially on the Mountain Pass theorem and on Ekeland s Variational principle. Copyright 2006 Mihai Mihailescu. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The goal of this paper is to study a nonlinear elliptic equation in which the divergence form operator - div a x v is involved. Such operators appear in many nonlinear diffusion problems in particular in the mathematical modeling of non-Newtonian fluids see 5 for a discussion of some physical background . Particularly the p-Laplacian operator - div Vu p-2Vu is a special case of the operator - div a x Vu . Problems involving the p-Laplacian operator have been intensively studied in the last decades. We just remember the work on that topic of João Marcos B. do O 7 Pfluger 12 Radulescu and Smets 14 and the references therein. In the case of more general types of operators we point out the papers of Joao Marcos B. do O 6 and Napoli and Mariani 4 . On the other hand when the operator - div a x Vu is of degenerate type we refer to Cirstea and Radulescu 15 and Motreanu and Radulescu 11 . In this paper we study the existence and multiplicity of non-trivial weak solutions to equations of the type - div a x Vu x u x G RN where the operator div a x Vu is nonlinear and can be also degenerate N 3 and function x u satisfies several hypotheses. Our goal is to show how variational techniques based on .
