Singular Elliptic Equations with Variable Exponents - Abdelaziz, Hellal (International Journal of Mathematics And its Applications, 2023)
This paper deals with study of the nonlinear singular elliptic equations in a bounded domain ΩR^N, (N≥ 2) with Lipschitz boundary Ω: \[ Au=fu^γ(·) +μ, \] Where A:=-div(a(·,Du)) is a Leray-Lions type operator which maps continuously W^1,p(·)_0(Ω) into its dual W^-1,p'(·)(Ω) whose simplest model is the p(·)-laplacian type operator (i.e. a(·,ξ)=|ξ|^p(·)-2ξ) such that f is a nonnegative function belonging to the Lebesgue space with variable exponents L^m(·)(Ω) with m(·) being small (or L^1(Ω)) and μ is a nonnegative function belongs to L^1(Ω) as nonhomogeneous datum while m:Ω→ (1,+∞), γ:Ω→ (0,1) are continuous functions satisfying certain conditions depend on p(·). We prove the existence, uniqueness, and regularity of nonnegative weak solutions for this class of problems with p(·)-growth conditions. More precisely, we will discuss that the nonlinear singular term has some regularizing effects on the solutions of the problem which depends on the summability of f, m(·), and the value of γ(·). The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents. Our results can be seen as a generalization of some results given in the constant exponents case.