Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 640841, 10 pages
doi:10.1155/2010/640841
Research Article

Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

Chunmei Yuan, Shujuan Guo, and Kaiyu Tong

College of Physics and Mathematics, Changzhou University, Changzhou, Jiangsu 213164, China

Received 8 October 2010; Revised 13 December 2010; Accepted 13 December 2010

Academic Editor: Jyh Horng Chou

Copyright © 2010 Chunmei Yuan 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.

Abstract

This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities − Δ 𝑢 = 𝜆 𝑎 ( 𝑥 ) 𝑢 𝑝 + 𝑏 ( 𝑥 ) 𝑢 𝑞 in Ω , 𝑢 > 0 in Ω , 𝑢 = 0 on 𝜕 Ω , where 𝜆 > 0 is a real parameter, 0 < 𝑝 < 1 < 𝑞 . Ω is a bounded domain in ℝ 𝑁 ( 𝑁 ≥ 3 ) , and 𝑎 ( 𝑥 ) and 𝑏 ( 𝑥 ) are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.