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Báo cáo hóa học: " Research Article On the Solvability of Superlinear and Nonhomogeneous Quasilinear Equations"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On the Solvability of Superlinear and Nonhomogeneous Quasilinear Equations | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 156063 10 pages doi 10.1155 2009 156063 Research Article On the Solvability of Superlinear and Nonhomogeneous Quasilinear Equations Gao Jia Qing Zhao and Chun-yan Dai College of Science University of Shanghai for Science and Technology Shanghai 200093 China Correspondence should be addressed to Gao Jia gaojia79@yahoo.com.cn Received 14 January 2009 Accepted 6 September 2009 Recommended by Michel C. Chipot Using Mountain Pass Lemma we obtain the existence of nontrivial weak solutions for a class of superlinear and nonhomogeneous quasilinear equations. The key factor in this paper is to use the new idea of near p-homogeneity in conjunction with variational techniques to obtain a new multiplicity result for a vast set of nonlinear equations such as the mean curvature equation and so on. Copyright 2009 Gao Jia et al. 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 In this paper we consider the nonlinear elliptic equation Qu x u p 2u f x u in Q J 1.1 u 0 on dQ Qu -1A DaAa x ịm uỴ 1.2 a m where Q c RN is a bounded open connected set N 1. Q is a quasilinear elliptic operator generalizing the p-Laplace that is Qu - div Vu p-2Vu . It appears that certain nonlinear mathematical models lead to nonlinear differential equations one of them describes the behavior compressible fluid in a homogeneous isotropic rigid porous medium such as the p-Laplace equation. And some purely mathematical properties of the p-Laplace seem to be a challenge for nonlinear analysis and their study leads to the development of new methods and approaches. 2 Boundary Value Problems These statements generalize certain homogeneous operators to a class of nonhomo-geneous quasilinear elliptic operators. In particular we get an equation involving the mean .