Volume 13, Issue 1

We prove the existence of solutions to three-fold symmetric elliptic systems in ℝ 2 which have six-fold symmetry, asymptotically approaching each of three minima of the potential as |x| → ∞ in two antipodal sectors of angle π/3.

Boundary value problems of Sturm-Liouville and periodic type for the second order nonlinear ODE uʺ + λf(t, u) = 0 are considered. Multiplicity results are obtained, for λ positive and large, under suitable growth restrictions on f(t, u) of superlinear type at u = 0 and of sublinear type at u = ∞. Only one-sided growth conditions are required. Applications are given to the equation uʺ + λq(t)f(u) = 0, allowing also a weight function q(t) of nonconstant sign.

We consider multi-valued elliptic variational inequalities for operators of the form u ↦ Au + ∂ 2 ψ(u, u), where A is a second order elliptic operator of Leray-Lions type, and u ↦ ∂ 2 ψ(u, u) is a multi-valued lower order term that may neither be lower nor upper semicontinuous, see e.g., Figure 2. More precisely, the lower order term is generated by some function (r, s) → ψ(r, s) with ψ : ℝ × ℝ → ℝ that is locally Lipschitz with respect to its second variable s for each r ∈ ℝ, and ∂ 2 ψ(r, s) denotes Clarke’s generalized gradient of s ↦ ψ(r, s). The novelty of this paper is that the multifunction r ↦ ∂ 2 ψ(r, s) may discontinuously depend on r in a certain specified way, which gives rise to the new class of discontinuous multi-valued lower order terms s ↦ ∂ 2 ψ(s, s). Though rich in structure, the characteristic features of this new class of multi-valued lower order terms can easily be described and verified. Our main goal is to provide an analytical framework and to prove existence and comparison results for this new class of multi-valued variational inequalities that includes the theory of variational-hemivariational inequalities as special case.

We investigate minimization and maximization of the principal eigenvalue of the Laplacian under Dirichlet boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, such optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. The question may have practical importance in the context of reserve design or pest control.

We consider the resonant-superlinear elliptic problem -Δu = λ 1 u + (u + ) q + f (x), with Dirichlet boundary conditions on a bounded regular domain of ℝ N . We assume that with s > N satisfies ∫ Ω fφ 1 < 0 and (λ 1 , φ 1 ) is the first eigenpair of −Δ on H 1 0 (Ω). We apply a non-well ordered lower and upper solution result on a family of modified problems and obtain a sequence of localized solutions of these modified problems. Thanks to this localization and a precise bootstrap argument we are abble to prove that for large modification, these solutions are, in fact, solutions of our initial problem. The problem was already considered in [2] by a totally different approach.

We consider quasilinear elliptic problems of the form Δ p u + f(u) = 0 over the half-space H = {x ∈ ℝ N : x 1 > 0} and over the quarter-space Q = {x ∈ ℝ N : x 1 > 0, x N > 0}. In the half-space case we assume u ≥ 0 on ∂H, and in the quarter-space case we assume that u ≥ 0 on {x 1 = 0} and u = 0 on {x N = 0}. Let u ≢ 0 be a bounded nonnegative solution. For some general classes of nonlinearities f , we show that, in the half-space case, lim x1→∞ u(x 1 , x 2 , ..., x N ) always exists and is a positive zero of f ; and in the quarter-space case, where V is a solution of the one-dimensional problem Δ p V + f(V) = 0 in ℝ + , V(0) = 0, V(t) > 0 for t > 0, V(+∞) = z, with z a positive zero of f . Our results extend most of those in the recent paper of Efendiev and Hamel [6] for the special case p = 2 to the general case p > 1. Moreover, by making use of a sharper Liouville type theorem, some of the results in [6] are improved. To overcome the difficulty of the lack of a strong comparison principle for p-Laplacian problems, we employ a weak sweeping principle.

We consider the principal eigenvalue of a cooperative system of elliptic boundary value problems as a parameter tends to infinity. The main aim is to introduce a new approach to deal with the limit problem by focusing on the resolvent operator corresponding to the system rather than the eigenvalue problem itself. This allows the consistent use of elementary properties of bilinear forms and the semi-groups they induce. At the same time we weaken assumptions in related work.

We consider radial solutions of an elliptic equation involving the p-Laplace operator and prove by a shooting method the existence of compactly supported solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane corresponding to an asymptotic Hamiltonian system and provides qualitative properties of the solutions.

We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restoring force has an asymptotically linear behavior. To this aim, after regularizing the problem, we use phase-plane analysis, and apply the Poincaré-Bohl fixed point Theorem to the associated Poincaré map, so to find a periodic solution of the regularized problem. Passing to the limit, we eventually find the “bouncing solution” we are looking for.

Motivated by the mean-value property characterizing harmonic functions and recently established asymptotic statistical formulas characterizing p-harmonic functions, we consider the Dirichlet problem for a functional equation involving a convex combination of the mean and median. We show that this problem has a continuous solution when it has both a subsolution and a supersolution. We demonstrate that solutions of these problems approximate p-harmonic functions and discuss connections with related results of Manfredi, Parviainen and Rossi.

We give an elementary proof of Poincaré-Miranda’s theorem based upon basic exterior calculus, and apply it to prove its geometric version due to Hurewicz. We state and prove a version of Poincaré-Miranda’s theorem on the Hilbert’s cube in l 2 .

Many known properties of the forced pendulum equation are generic. This means that they hold on a residual set of forcings. The classical duality category/measure inspires the main question in the paper: are these properties also valid for a prevalent set of forcings? This is the case for a key property: the non-degeneracy of periodic solutions. The main tool in the proof is a prevalent version of the parametric transversality theorem.

We consider quasilinear parabolic evolution equations in the situation where the set of equilibria forms a finite-dimensional C 1 -manifold which is normally hyperbolic. The existence of foliations of the stable and unstable manifolds is shown assuming merely C 1 -regularity of the underlying equation.

We study local and global bifurcations of the following elliptic system where the constant a is greater than the principal eigenvalue of (-Δ,Ω), and Ω ⊂ ℝ N is a bounded smooth domain (N ≤ 3). The parameters μ 1 ,μ 2 are real numbers. We identify some intervals of β in which there exist finitely many bifurcation points with respect to a trivial solution branch. In particular, when N = 1 or Ω is radial, every bifurcation branch exists globally and is characterized by a nodal property for the difference of the two components of the solutions.