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

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.

Keywords

• p(x)-Laplacian
• singular nonlinear term
• sub-supersolution method