Volume 11, Issue 1

This paper is about the existence and some properties of solutions of variational inequalities associated with the 2nd order inclusion div[A(x, ∇u)] + L ∈ f (x, u) in Ω, where the lower order term f (x, u) is a general multivalued function. Both coercive and noncoercive cases are considered. In the noncoercive case, we use a sub-supersolution approach to study the existence, comparison, and other properties of the solution set such as its compactness, directedness, and the existence of extremal solutions.

In this paper we consider the problem of minimizing the functional We prove Lipschitz continuity for each minimizer u and establish the nondegeneracy at the free boundary (∂[u > 0]) ∩ Ω and the locally uniform positive density of the sets [u > 0] and [u = 0]. In particular we obtain that the Lebesgue measure of the free boundary is zero.

The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of [4], the existence of at least three nontrivial solutions to the quasilinear elliptic equation −Δ p(x) u = |u| q(x)−2 u + λ f (x, u) in a smooth bounded domain Ω of R N with homogeneous Dirichlet boundary conditions on ∂Ω. We assume that {q(x) = p∗(x)} ≠ ø, where p∗(x) = Np(x)/(N − p(x)) is the critical Sobolev exponent for variable exponents and Δ p(x) u = div(|∇u| p(x)−2 ∇u) is the p(x)−laplacian. The proof is based on variational arguments and the extension of concentration compactness method for variable exponent spaces.

We prove the existence of infinitely many subharmonic solutions, with prescribed nodal properties, for a planar Hamiltonian system Jz′ = Δ z H(t, z), with H periodic in the first variable. The goal is achieved by performing estimates of the rotation numbers with respect to deformed polar coordinates and applying Ding’s version of the Poincaré-Birkhoff fixed point theorem.

In this paper we use functions on T n−1 with n ≥ 2 possessing minimal number of critical points given by Ljusternik-Schnirelman theory to construct spatially n-periodic Hamiltonian systems which possess precisely n distinct rotation type solutions with prescribed energy and rotation type vector.

Given a space X, we study the homotopy type of B n (X) the space obtained as the “union of all (n−1)-simplexes spanned by points in X” or the space of “formal barycenters of weight n or less” of X. This is a space encountered in non-linear analysis under the name of space of barycenters or in differential geometry in the case n = 2 as the space of chords. We first relate this space to a more familiar symmetric join construction and then determine its stable homotopy type in terms of the symmetric products on the suspension of X. This leads to a complete understanding of the homology of B n (X) as a functor of X, and to an expression for its Euler characteristic given in terms of X. A sharp connectivity theorem is also established. Finally the case of spheres S is studied in details and the homotopy type of B n (S ) is described, generalizing in this way an early and beautiful result of James, Thomas, Toda and Whitehead.

We consider an exotic contact form α on S 3 and we establish explicitly the existence of a non singular vector field v in ker(α) such that the non-singular one-differential form β(・) := dα(v, ・) is a contact form on S 3 with the same orientation as α. In particular this means that a Legendre transform can be completed.

We establish existence results for solutions to functional boundary value problems for φ- Laplacian ordinary differential equations assuming there are lower and upper solutions and Lipschitz bounding surfaces for the derivative which we adapt to our problem. Our results apply to some problems which do not satisfy Nagumo growth bounds. Moreover they contain as special cases many results for the p- and ɸ-Laplacians as well as many results where the boundary conditions depend on n-points or even functionals. Our boundary conditions generalize those of Fabry and Habets, Cabada and Pouso, Cabada, O’Regan and Pouso, and many others.

This paper is concerned with existence and uniqueness of solutions for a doubly nonlinear degenerate parabolic problem of the type β(w) t − div a(x, Dw) + ∂ j ( ., β(w) ) ∋ f, where α is a Leray-Lions operator, β is a nondecreasing continuous function and ∂ j(., r) is a maximal monotone graph with respect to r defined on a closed interval of ℝ. Particular cases of j correspond to the so called obstacle problem.

Motivated by the classical Coulomb central motion model, we study the existence of T-periodic solutions for the nonlinear second order system of singular ordinary differential equations u′′ + g(u) = p(t). Using topological degree methods, we prove that when the nonlinearity g : ℝ N \{0} →ℝ N is continuous, repulsive at the origin and bounded at infinity, if an appropriate Nirenberg type condition holds then either the problem has a classical solution, or else there exists a family of solutions of perturbed problems that converge uniformly and weakly in H 1 to some limit function u. Furthermore, under appropriate conditions we prove that u is a classical solution.

In this article we consider the following problem with max{p − 1, 1} ≤ q ≤ p < N, Ω a bounded domain and f, g positive measurable functions in Ω. We give assumptions on g with respect to q and p for which for all λ > 0 and all f ∈ L 1 (Ω), f ≥ 0, problem (P) has a positive solution. In particular, we focus our attention on the case when 0 ∈ Ω and to prove that the assumptions on g are optimal. Furthermore, regularity results are given.