A Singular Semilinear Elliptic Equation with a Variable Exponent

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 . -Δ u=f(x)uγ(x) Ω, u=0% on ∂Ω. Here Ω is an open bounded set of ℝ N RN , γ ⁢ ( x ) γ(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 ) H¹₀(Ω) when γ ⁢ ( x ) ≤ 1 γ(x)≤ 1 in a strip around the boundary. For another case, we prove that the solution belongs to H loc 1