International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 109214, 21 pages
http://dx.doi.org/10.1155/2012/109214
Research Article

Multiple Positive Solutions for a Quasilinear Elliptic System Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Tsing-San Hsu

Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan

Received 20 April 2011; Accepted 20 December 2011

Academic Editor: Marco Squassina

Copyright © 2012 Tsing-San Hsu. 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.

Abstract

Let Ω ∋ 0 be an-open bounded domain in ℝ 𝑁 ( 𝑁 ≥ 3 ) and 𝑝 ∗ = ( 𝑝 𝑁 / ( 𝑁 − 𝑝 ) ) . We consider the following quasilinear elliptic system of two equations in 𝑊 0 1 , 𝑝 ( Ω ) × 𝑊 0 1 , 𝑝 ( Ω ) : − Δ 𝑝 𝑢 = 𝜆 𝑓 ( 𝑥 ) | 𝑢 | 𝑞 − 2 𝑢 + ( 𝛼 / ( 𝛼 + 𝛽 ) ) ℎ ( 𝑥 ) | 𝑢 | 𝛼 − 2 𝑢 | 𝑣 | 𝛽 , − Δ 𝑝 𝑣 = 𝜇 𝑔 ( 𝑥 ) | 𝑣 | 𝑞 − 2 𝑣 + ( 𝛽 / ( 𝛼 + 𝛽 ) ) ℎ ( 𝑥 ) | 𝑢 | 𝛼 | 𝑣 | 𝛽 − 2 𝑣 , where 𝜆 , 𝜇 > 0 , Δ 𝑝 denotes the 𝑝 -Laplacian operator, 1 ≤ 𝑞 < 𝑝 < 𝑁 , 𝛼 , 𝛽 > 1 satisfy 𝑝 < 𝛼 + 𝛽 ≤ 𝑝 ∗ , and 𝑓 , 𝑔 , ℎ are continuous functions on Ω which are somewhere positive but which may change sign on Ω . We establish the existence and multiplicity results of positive solutions to (the above mentioned quasilinear elliptic system equations) by variational methods.