The problem −Δu + F(V (εx), u) = 0 is considered in Rn. For small ε > 0, solutions are obtained that approach, as ε → 0, a linear combination of specified functions, mutually translated by O(1/ε). These are the so-called multi-bump solutions. The method involves a rescaling of the variables and the use of a modified implicit function theorem. The usual implicit function theorem is inapplicable, owing to lack of convergence of the derivative of the nonlinear Hilbert space operator, obtained after an appropriate rescaling, in the operator-norm topology. An asymptotic formula for the solution for small ε is obtained.

We study the contact between nonlinearly elastic bodies by variational methods. After the formulation of the mechanical problem, we provide existence results based on polyconvexity and on quasiconvexity. We then derive the Euler—Lagrange equation as a necessary condition for minimizers. Here Clarke's generalized gradients are an essential tool for treating the nonsmooth obstacle condi

Let f be transcendental and meromorphic in the plane and let the non-homogeneous linear differential polynomials F and G be defined by where k,n ∈ N and a, b and the aj, bj are rational functions. Under the assumption that F and G have few zeros, it is shown that either F and G reduce to homogeneous linear differential polynomials in f + c, where c is a rational function that may be computed explicitly, or f has a representation as a rational function in solutions of certain associated linear differential equations, which again may be determined explicitly from the aj, bj and a and b.

This paper is concerned with multi-dimensional non-isentropic Euler–Poisson equations for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyse the quasi-neutral limit for Cauchy problems with prepared initial data. It is shown that the small-parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions. Moreover, the formal limit is justified.

Under the assumption that the Radon measure μ on Rd satisfies only some growth condition, the authors prove that, for the maximal singular integral operator associated with a singular integral whose kernel only satisfies a standard size condition and the Hörmander condition, its boundedness in Lebesgue spaces Lp(μ) for any p ∈ (1, ∞) is equivalent to its boundedness from L1(μ) into weak L1(μ). As an application, the authors verify that if the truncated singular integral operators are bounded on L2(μ) uniformly, then the associated maximal singular integral operator is also bounded on Lp(μ) for any p ∈ (1, ∞).

We prove that the reflection coefficient of one-dimensional Schrödinger operators with potentials supported on a half-line can be represented in the upper half-plane as the quotient of a contractive analytic function and a properly regularized Blaschke product. We apply this fact to obtain a new trace formula and trace inequality for the reflection coefficient that yields a description of the Weyl m-function of Dirichlet half-line Schrödinger operators with slowly decaying potentials q subject to Among others, we also refine the 3/2-Lieb-Thirring inequality.

We study the existence and multiplicity of positive solutions of the non-homogeneous elliptic equation where N ≥ 3, the nonlinearity f is superlinear at both zero and infinity, q is a non-trivial, non-negative function, and a and b are non-negative parameters. A typical model is given by f(u) = up, with p ≥ 1.

Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

We denote the group of homotopy set [X, U(n)] by the unstable K1-group of X. In this paper, using the unstable K1-group of the multi-suspended CP2, we give a necessary condition for two principal SU(n)-bundles over §4 to have the associated gauge group of the same homotopy type, which is an improvement of the result of Sutherland and, particularly, show the complete classification of homotopy types of SU(3)-gauge groups over S4.

For a class of nonlinear parabolic equations and inequalities, we establish conditions guaranteeing that any solution tends to 0 as t → ∞.

The equation considered in this paper is tp(φp(x′))′ + g(x) = 0, where φp(x′) = |x′|p−2x′ with p > 1, and g(x) satisfies the signum condition xg(x) > 0 if x ≠ 0 but is not assumed to be monotone. Our main objective is to establish a criterion on g(x) for all non-trivial solutions to be non-oscillatory. The criterion is the best possible. The method used here is the phase-plane analysis of a system equivalent to this differential equation. The asymptotic behaviour is also examined in detail for eventually positive solutions of a certain half-linear differential equation.

Let Ω be a bounded Lipschitz domain in R2, let H be a 2 × 2 diagonal matrix with det(H) = 1. Let ε > 0 and consider the functional over AF ∩ W2,1(Ω), where AF is the class of functions from Ω satisfying affine boundary condition F. It can be shown by convex integration that there exists F ∉ SO(2) ∪ SO(2)H and u ∈ AF with I0(u) = 0. Let 0 < ζ1 < 1 < ζ2 < ∞, .

In this paper we begin the study of the asymptotics of mε ≔ infBF∩W2,1Iε for such F. This is one of the simplest minimization problems involving surface energy for which we can hope to see the effects of convex integration solutions. The only known lower bounds are lim infε→0mε/ε = ∞.

We link the behaviour of mε to the minimum of I0 over a suitable class of piecewise affine functions. Let {τi} be a triangulation of Ω by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation {τi}. For the function u ∈ BF let be the interpolant, i.e. the function we obtain by defining ũ⌊τi to be the affine interpolation of u on the corners of τi. We show that if for some small ω > 0 there exists u ∈ BF ∩ W2,1 with then, for h = ε(1+6399ω)/3201, the interpolant satisfies I0(ũ) ≤ h1−cω.

Note that it is trivial that , so we reduce the problem of non-trivial (scaling) lower bounds on mε/ε to the problem of non-trivial lower bounds on .

We consider elliptic systems with discontinuous coefficients and prove that if the known term belongs to the Morrey space Lp,λ, then the highest-order derivatives of the local solution belong to the same space. We also obtain local Hölder continuity for lower-order derivatives.

Towards the end of the nineteenth century, Halphen studied a remarkable sequence of higher-order linear equations with doubly periodic coefficients, generalizations of a certain Lamé equation, having the property that quotients of solutions are single valued. Here we consider further generalizations where, instead of the Weierstrass ℘-function, the coefficients depend on the first Painlevé transcendent. Using these equations, we obtain new higher-order systems of nonlinear equations having the Painlevé property. We also give new results on the interpretation of the Painlevé tests with regard to the representations of solutions, general and particular, afforded by various branches, and to understanding the corresponding pattern of compatibility conditions.

We give a positive answer to an open problem about Hardy's inequality raised by Brézis and Vázquez, and another result obtained improves that of Vázquez and Zuazua. Furthermore, by this improved inequality and the critical-point theory, in a k-order Sobolev–Hardy space, we obtain the existence of multi-solution to a nonlinear elliptic equation with critical potential and critical parameter.

We consider an overdetermined system of elliptic partial differential equations arising in the Navier–Stokes equations. This analysis enables us to prove that the well-known classical solutions such as Couette flows and others are the only solutions that satisfy both the stationary Navier–Stokes and Euler equations.

In this paper we analyse the methodology of the theory of differential inclusions. First, we emphasize that any sequence of piecewise affine functions with successive elements obtained by perturbations of preceding functions in the sets of their affinity converges strongly, together with the gradients. This gives a simple algorithm with which to construct sequences of approximate solutions that converge to exact solutions (neither the specific choice suggested by ‘the method of convex integration for Lipschitz functions’ nor Baire category methodology is required). We then suggest a functional that is defined in the set of admissible functions and measures maximal oscillations produced by sequences of admissible functions weakly convergent to a given function. This functional can be used to prove that the set of stable solutions is dense in the weak topology in the closure of the set of admissible functions either via the Baire category lemma or via a specific choice of strictly convergent sequences.

We explain how the above-mentioned methods of finding solutions to differential inclusions are connected to earlier results on weak–strong convergence, i.e. to results on stability, in the calculus of variations and in differential inclusions.

We also include information on developments in the subject in the three years after the results of this work were obtained.

The Cauchy problem for semi-linear heat equations with singular initial data is studied, where N > 2, p > (N + 2)/N, and l > 0 is a parameter. We establish the existence and multiplicity of positive self-similar solutions for the problem by applying the ordinary differential equation shooting method to the corresponding spatial profile problem.

The full nonlinear dynamic von Kárm´n system depending on a small parameter ε > 0 is considered. We study the asymptotic behaviour of the total energy associated with the model for large t and ε → 0. Introducing appropriate boundary feedback, we show that the total energy of a solution of the corresponding damped model decays exponentially as t → +∞, uniformly with respect to the parameter ε > 0. As ε → 0, we obtain a damped plate model for which the energy also tends to zero exponentially. The limit system can be viewed as new variant of the so-called Timoshenko model. It consists of a second-order hyperbolic equation for transversal vibrations of the plate coupled with a first-order ordinary differential equation whose solution appears as coefficient of the plate model and takes into account (when ε → 0) the contribution of the tangential components.

The existence of solutions of the Dirichlet problems for the prescribed mean-curvature equation on some unbounded domains in Rn(n ≥ 2) is proved. The results are proved using a modified version of the Perron method, where a subsolution is a solution to the minimal surface equation, while a supersolution is not constructed; instead, the role played by a supersolution is replaced by the estimates on the uniform bounds on the liftings of subfunctions on compact sets.