Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài :Multiple positive solutions for a class of quasilinear elliptic equations involving concave-convex nonlinearities and Hardy terms | Hsu Boundary Value Problems 2011 2011 37 http content 2011 1 37 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms Tsing-San Hsu Correspondence tshsu@. Center for General Education Chang Gung University Kwei-San Tao-Yuan 333 Taiwan ROC Abstract In this paper we are concerned with the following quasilinear elliptic equation u p 2u x p kf x u q 2u g x u p 2u in u 0 on 90 p where o c RN is a smooth domain with smooth boundary cO such that 0 e o Apu dV Vu p-2Vu 1 p N p p N2- p l 0 1 q p sign-changing weight functions f and g are continuous functions on p Npp p is the best Hardy constant and p N p is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold the multiplicity of positive solutions to this equation is verified. Keywords Multiple positive solutions critical Sobolev exponent concave-convex Hardy terms sign-changing weights 1 Introduction and main results Let o be a smooth domain not necessarily bounded in RN N 3 with smooth boundary do such that 0 e o. We will study the multiplicity of positive solutions for the following quasilinear elliptic equation u p-2u Apu p kf x u q 2u g x u p 2u in u 0 x p on where Apu divdVuf- Vu 1 p N p p Npp p p is the best Hardy constant l 0 1 q p p N p is the critical Sobolev exponent and the weight functions f g ÍÃ R are continuous which change sign on o. Let D0 p be the completion of c with respect to the norm fa Vufdx 1 1 . The energy functional of is defined on D0 p by Jk u 1 f Vu p p dx k f u qdx 1 g u p dx. p Q x p q Q p Q Then Jk e C1 D0 p R . u e D0 p 0 is said to be a solution of if J k u v 0 for all v e D1 p O and a solution of is a critical point of Ji. Springer 2011 Hsu licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
