Structural stability for variable exponent elliptic problems, I: The p(x)p(x)-Laplacian kind problems

We study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form b(un)−divan(x,∇un)=fn. The equation is set in a bounded domain Ω of RNRN and supplied with the homogeneous Dirichlet boundary condition on ∂Ω. Here bb is a non-decreasing function on RR, and (an(x,ξ))n(an(x,ξ))n is a family of applications which verifies the classical Leray–Lions hypotheses but with a variable summability exponent pn(x)pn(x), 1