This paper investigates the existence of solutions for boundary variational–hemivariational inequalities of elliptic type at resonance as well as at non-resonance. Using the notion of the generalized gradient of Clarke and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of variational–hemivariational inequalities of elliptic type.

We prove the existence of a very singular solution to when 1 < p < (N + 2)/(N + 1).

For a wide class of nonlinearities f(u) satisfying but not necessarily Lipschitz continuous, we study the quasi-linear equation where T = {x = (x1, x2, …, xN) ∈ RN: x1 > 0} with N ≥ 2. By using a new approach based on the weak maximum principle, we show that any positive solution on T must be a function of x1 only. Under our assumptions, the strong maximum principle does not hold in general and the solution may develop a flat core; our symmetry result allows an easy and precise determination of the flat core.

In 1992, Fountain and Lewin showed that any proper ideal of an endomorphism monoid of a finite independence algebra is generated by idempotents. Here the ranks and idempotent ranks of these ideals are determined. In particular, it is shown that when the algebra has dimension greater than or equal to three the idempotent rank equals the rank.

Asymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.

We investigate different types of periodic solutions of a fourth-order, nonlinear differential equation, which has been proposed as a model for travelling waves in suspension bridges. We develop a shooting argument, which enables us to prove the existence of two families of multibump periodic solutions, each containing a countably infinite number of different solutions

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the C0-norm of the initial data is sufficiently small, while the C1-norm of the initial data can be large. The analysis is based on a priori estimates, which are obtained by a generalised Lax transformation.

We study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

In this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem −Δu + u = up + ƒ(z) in Ω has at least two positive solutions, where

a bounded C1,1 domain S = × ω Rn, D is a bounded C1,1 domain in Rm+n such that D ⊂⊂ S and Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing that D is small.

We consider a quasilinear elliptic problem of the form

where λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

By homotopy linear algebra we mean the study of linear functors between slices of the ∞-category of ∞-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into ∞-categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality à la Baez, Hoffnung and Walker compatible with this duality. We needed these results to support our work on incidence algebras and Möbius inversion over ∞-groupoids; we hope that they can also be of independent interest.

In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form Aε = −div(aε (x, ∇u)), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G-converge to an operator Ahom = −div(ahom(x, ∇u)). We show that any limit point λ of a sequence of eigenvalues λε is an eigenvalue of the limit operator Ahom, where λε is an eigenvalue corresponding to the operator Aε. We also show the convergence of the sequence of first eigenvalues to the corresponding first eigenvalue of the homogenized operator.

Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional (3D) fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier–Stokes equations, is a mathematical analogue of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier–Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme.

In this paper we apply a theorem of Khelemskiĭ and Sheĭnberg, characterising amenability by means of bounded approximate identities, to weighted discrete convolution algebras. In doing this we give a condition for a weighted discrete convolution algebra to have a bounded approximate identity. Under the condition that the semigroup (S,.) is one-sided cancellative, we prove that, if some weighted discrete convolution algebra on S is amenable, then (S,.) is actually a group. We further characterise all amenable weighted discrete convolution algebras on groups, thus extending a well-known theorem of B. E. Johnson [9].

We prove the well-posedness for a one-dimensional free boundary problem arising from some reaction diffusion system. The interfacial point hits a boundary point in finite time or remains inside for all time. In the large diffusion limit, the system is reduced to ordinary differential equations of finite dimension.

The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.

We use methods of bifurcation theory to study properties of solution curves for a class of quasilinear two-point problems. Unlike semilinear equations, here, solution curves may stop at some point, or solution curves may turn the ‘wrong way’ (compared with semilinear equations) as in a paper by Habets and Omari, where the prescribed mean curvature equation was considered. This class of equations will be our main example. Another difference from semilinear equations is that the bifurcation diagram may depend on the length of the interval, as was discovered recently by Pan, who considered the prescribed mean curvature equation and f(u) = eu. We generalize this result to convex f(u), with f(0) > 0, and to more general quasilinear equations. We also give formulae which allow us to compute all possible turning points and the direction of the turn, generalizing similar formulae in Korman et al. We also present a numerical computation of the bifurcation curves.

We derive geometrical information on smooth surfaces in $\mathbb{R}^4$ from the geometry of their images under linear projections to 3-spaces.

We study the initial value problem for the nonlinear Schrödinger equation

Under suitable regularity assumptions on f and ø and growth and sign conditions on f, it is shown that the maximum norms of solutions to (*) decay as t→² ∞ at the same rate as that of solutions to the free Schrödinger equation.