Electron. J. Diff. Equ., Vol. 2014 (2014), No. 155, pp. 1-21.

Existence of positive solutions for p(x)-Laplacian equations with a singular nonlinear term

Jingjing Liu, Qihu Zhang, Chunshan Zhao

Abstract:
In this article, we study the existence of positive solutions for the p(x)-Laplacian Dirichlet problem
$$ -\Delta _{p(x)}u=\lambda f(x,u) $$
in a bounded domain $\Omega \subset \mathbb{R}^{N}$. The singular nonlinearity term f is allowed to be either $f(x,s)\to +\infty $, or $f(x,s)\to +\infty $ as $s\to 0^{+}$ for each $x\in \Omega $. Our main results generalize the results in [15] from constant exponents to variable exponents. In particular, we give the asymptotic behavior of solutions of a simpler equation which is useful for finding supersolutions of differential equations with variable exponents, which is of independent interest.

Submitted July 2, 2013. Published July 7, 2014.
Math Subject Classifications: 35J25, 35J65, 35J70.
Key Words: p(x)-Laplacian; singular nonlinear term; sub-supersolution method.

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Jingjing Liu
College of Mathematics and Information Science
Zhengzhou University of Light Industry
Zhengzhou, Henan 450002, China
email: jingjing830306@163.com
Qihu Zhang (corresponding author)
College of Mathematics and Information Science
Zhengzhou University of Light Industry
Zhengzhou, Henan 450002, China
email:zhangqihu@yahoo.com, zhangqh1999@yahoo.com.cn
Chunshan Zhao
Department of Mathematical Sciences
Georgia Southern University
Statesboro, GA 30460, USA
email: czhao@GeorgiaSouthern.edu

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