Global $$C^{1,\al pha }$$ regularity and e xistence of multi ple solutions for singular p( x)-La placian equations

In this paper we study the following singular p(x)-Laplacian problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \text{ div } \left( |\nabla u|^{p(x)-2} \nabla u\right) =\frac{ \lambda }{u^{\beta (x)}}+u^{q(x)}, &{} \text{ in }\quad \Omega , \\ u>0, &{} \text{ in }\quad \Omega , \\ u=0, &{} \text{ on }\quad \partial \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 2\), with smooth boundary \(\partial \Omega \), \(\beta \in C^1(\bar{\Omega })\) with \( 0< \beta (x) <1\), \(p\in C^1(\bar{\Omega })\), \(q \in C(\bar{\Omega })\) with \(p(x)>1\), \(p(x)< q(x) +1 <p^*(x)\) for \(x \in \bar{\Omega }\), where \( p^*(x)= \frac{Np(x)}{N-p(x)} \) for \(p(x) <N\) and \( p^*(x)= \infty \) for \( p(x) \ge N\). We establish \(C^{1,\alpha }\) regularity of weak solutions of the problem and strong comparison principle. Based on these two results, we prove the existence of multiple (at least two) positive solutions for a certain range of \(\lambda \).