Abstract In this paper we consider singular semilinear elliptic equations with a variable exponent whose model problem is - Δ ⁢ u = f ⁢ ( x ) u γ ⁢ ( x )   in ⁢ Ω , u = 0   on ⁢ ∂ ⁡ Ω . $-\Delta u=\frac{f(x)}{u^{\gamma(x)}}\quad\text{in }\Omega,\qquad u=0\quad\text% {on }\partial\Omega.$ Here Ω is an open bounded set of ℝ N ${\mathbb{R}^{N}}$ , γ ⁢ ( x ) ${\gamma(x)}$ is a positive continuous function and f ⁢ ( x ) ${f(x)}$ is a positive function that belongs to a certain Lebesgue space. We prove that there exists a solution to this problem in the natural energy space H 0 1 ⁢ ( Ω ) ${H^{1}_{0}(\Omega)}$ when γ ⁢ ( x ) ≤ 1 ${\gamma(x)\leq 1}$ in a strip around the boundary. For another case, we prove that the solution belongs to H loc 1 ⁢ ( Ω ) ${H^{1}_{\mathrm{loc}}(\Omega)}$ and that it is zero on the boundary in a suitable sense.