Structural stability for nonlinear elliptic problems of the $p(x)-$ and $p(u)$-laplacian kind
Boris Andreïanov, Mostafa Bendahmane, Stanislas Ouaro · 2009
Abstract. This work consists of two parts. In the first one, we prove the structural stability (i.e., the continuous dependence on the coefficients) of solutions of the elliptic problems under the form b(un) − div an(x,∇un) = fn in a bounded domain Ω of R N with homogeneous Dirichlet boundary data on ∂Ω. Here b is a non-decreasing function on R, and an(x, ξ) n is a family of applications which verifies the classical Leray-Lions hypotheses but with a variable summability exponent pn(x) , 1 < p − ≤ pn(·) ≤ p+ < + ∞. The need for making vary p(x) arises, for instance, in the numerical analysis of the p(x) − laplacian problem. Uniqueness and existence for these problems are well understood by now. We prove a continuous dependence result for weak and renormalized solutions of this problem. Notice that, besides the interest of its own, the renormalized solutions ’ framework also permits to deduce optimal convergence results for the weak solutions of the problem. The technique presented in the first part permits to avoid the use of a fixed duality framework (like the W