Existence and regularity results for a class of equations with logarithmic growth

We study the existence and the regularity of the solutions of the following Dirichlet problem {−div(A(x,∇u))=finΩu=0on∂Ω where A(x,ξ)A(x,ξ) is slowly increasing at ∞∞ as |ξ|→+∞|ξ|→+∞. More precisely, we consider the model case A(x,ξ)=M(x)ξ|ξ|logα(1+|ξ|), where M(x)M(x) is a bounded elliptic matrix and α>0α>0 is a fixed exponent. We will show that if f∈Ln(Ω)f∈Ln(Ω) then there exists a unique solution u∈W01,1(Ω)∩L∞(Ω), such that ∇u∈LlogαL(Ω)∇u∈LlogαL(Ω). Moreover, we prove that there exists an exponent ε>0ε>0 such that |∇u|logα+ε(1+|∇u|)∈L1(Ω)|∇u|logα+ε(1+|∇u|)∈L1(Ω). In case the matrix M(x)M(x) is assumed to be Hölder continuous, we establish a higher differentiability result for the solution in the scale of fractional order Sobolev spaces which yields a higher integrability result in the scale of Lebesgue spaces.