Band 9, Heft 2

We consider a model describing compressible nuclear waste disposal contamination in porous media. The transport of brine and radionuclides is described by a nonlinear coupled degenerate parabolic system. The viscosity of the fluid is unbounded and concentrations and temperature dependent. We study the asymptotic behavior of the model for little Peclet numbers.

In this paper we prove the existence of constant mean curvature hypersurfaces which are cylindrically bounded and which bifurcate from the family of immersed constant mean curvature hypersurface of revolution. Based on the study of the spectrum of the Jacobi operator (the linearized mean curvature) about this family, the existence of new branches follows from a bifurcation result of Crandall and Rabinowitz.

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincaré T-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index.

We prove Brunn-Minkowski-type inequalities for functionals related to nontrivial weak solutions of Dirichlet boundary value problems for Monge-Ampère equations of the form det D 2 u = |u| P , where 0 ≤ p < n, in bounded strictly convex domains in ℝ n . The expected equality conditions are also established. These results generalize inequalities known for analogous problems for the p-Laplacian. The existence of solutions to these problems is an extension of known results for more regular domains, and is proved by approximation. The principal technique used in the proof of the main theorem is the same as that used to establish a similar inequality for the eigenvalue of the Monge-Ampère operator.

In this paper, we consider a singular elliptic system with both concave-convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

We prove a result of existence and localization of positive solutions of the Dirich- let problem for -Δ p u = w(x)f(u) in a bounded domain Ω, where Δ p is the p-Laplacian, w is a weight function and the nonlinearity f(u) satisfies certain local bounds. As in previous results in radially symmetric domains by two of the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on f. A positive solution is obtained by applying the Schauder Fixed Point Theorem and such approach allows us to construct ordered sub- and super-solutions for the problem, thus producing an iterative method to obtain a positive solution. Our result leads not only to asymptotic conditions on the nonlinearity which provides the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm, but also to an estimate of the first eigenvalue λ p (Ω, w) of the p-Laplacian operator with weight w. For w ≡ 1, we compare our lower bound for λ p (Ω, 1) with that obtained by means of the Cheeger constant h(Ω). We give a characterization of this constant in terms of the solution of the torsional creep problem -Δ pɸp = 1 in Ω with Dirichlet boundary data, which offers a good approximation of the first eigenvalue of the p-Laplacian for p near 1.

We study existence and regularity of the solutions for some anisotropic elliptic problems with homogeneous Dirichlet boundary conditions in bounded domains.

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness for solutions to Δu = e q(x)u in a bounded smooth domain Ω with the boundary condition u(x) → + ∞ as dist(x, ∂Ω) → 0. The exponent q(x) is assumed to be a Hölder continuous function which is either positive on ∂Ω or is positive in a neighborhood of ∂Ω maybe vanishing on ∂Ω. When dealing with nonnegative exponents q we are allowing nonempty interior regions Ω 0 ⊂ Ω where q vanishes. Changing sign exponents q will be also considered.

We prove uniqueness for the globally modified Navier-Stokes equations recently introduced by Caraballo, Real & Kloeden in [1] for initial conditions in the space H of square-summable divergence-free vector fields.