We study the structural stability (i.e., the continuous dependence on coefficients) of solutions of the elliptic problems under the form b(u n) − div a n (x,∇u n) = f n. The equation is set in a bounded domain Ω of R N and supplied with the homogeneous Dirichlet boundary condition on ∂Ω. Here b is a non-decreasing function on R, and a n (x, ξ) n is a family of applications which verifies the classical Leray-Lions hypotheses but with a variable summability exponent p n (x), 1 < p − ≤ p n (•) ≤ 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 apply the stability properties to further generalize the existence results. The continuous dependence result we prove is valid for weak and for renormalized solutions. Notice that, besides the interest of its own, the renormalized solutions' framework also permits to deduce optimal convergence results for the weak solutions. Our technique avoids the use of a fixed duality framework (like the W 1,p(x) 0 (Ω)-W −1,p ′ (x) (Ω) duality), and thus it is suitable for the study of problems where the summability exponent p also depends on the unknown solution itself, in a local or in a non-local way. The sequel of this paper will be concerned with wellposedness of some p(u)-laplacian kind problems and with existence of solutions to elliptic systems with variable, solution-dependent growth exponent.

We study well-posedness for elliptic problems under the form b(u) − div a(x, u,∇u) = f, where a satisfies the classical Leray-Lions assumptions with an exponent p that may depend both on the space variable x and on the unknown solution u. A prototype case is the equation u − div | ∇u| p(u)−2 ∇u = f. We have to assume that inf x∈Ω, z∈R p(x, z) is greater than the space dimension N. Then, under mild regularity assumptions on Ω and on the nonlinearities, we show that the associated solution operator is an order-preserving contraction in L 1 (Ω). In addition, existence analysis for a sample coupled system for unknowns (u, v) involving the p(v)-laplacian of u is carried out. Coupled elliptic systems with similar structure appear in applications, e.g. in modelling of stationary thermo-rheological fluids.

We study well-posedness of triply nonlinear degenerate ellipticparabolic-hyperbolic problems of the kindin a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities b, φ and ψ are supposed to be continuous non-decreasing, and the nonlinearityã falls within the Leray-Lions framework. Some restrictions are imposed on the dependence ofã(u, ∇φ(u)) on u and also on the set where φ degenerates. A model case isã(u, ∇φ(u)) =f(b(u), ψ(u), φ(u)) + k(u)a 0 (∇φ(u)), with a nonlinearity φ which is strictly increasing except on a locally finite number of segments, and the nonlinearity a 0 which is of the Leray-Lions kind. We are interested in existence, uniqueness and stability of L ∞ entropy solutions. For the parabolic-hyperbolic equation (b = Id), we obtain a general continuous dependence result on data u 0 , f and nonlinearities b, ψ, φ,ã.Similar result is shown for the degenerate elliptic problem, which corresponds to the case of b ≡ 0 and general non-decreasing surjective ψ. Existence, uniqueness and continuous dependence on data u 0 , f are shown in more generality. For instance, the assumptions [b + ψ](R) = R and the continuity of φ • [b + ψ] −1 permit to

In this paper, we prove the existence and uniqueness of weak solutions for a family of discrete boundary value problems for data f which belong to a discrete Hilbert space H. Moreover, as an extension, we prove some existence results of weak solutions for more general data f depending on the solution.

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