We prove the existence of radial solutions of concentrating on a sphere for potentials which might be zero and might decay to zero at infinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov–Schmidt reduction.

We characterize the weighted Hardy inequalities for monotone functions in In dimension n = 1, this recovers the standard theory of Bp weights. For n > 1, the result was previously only known for the case p = 1. In fact, our main theorem is proved in the more general setting of partly ordered measure spaces.

We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.

We prove that a (bounded, linear) operator acting on an infinite-dimensional, separable, complex Hilbert space can be written as a product of two quasi-nilpotent operators if and only if it is not a semi-Fredholm operator. This solves the problem posed by Fong and Sourour in 1984. We also consider some closely related questions. In particular, we show that an operator can be expressed as a product of two nilpotent operators if and only if its kernel and co-kernel are both infinite dimensional. This answers the question implicitly posed by Wu in 1989.

We prove the existence of multiple bound states of the nonlinear Schrödinger equation −Δu + V(x)u = f(u). Here the linear potential V is continuous and bounded from below, and the nonlinearity f is of asymptotically linear type. We show that, under certain assumptions on the spectrum of the Schrödinger operator −Δ + V and the asymptotic behaviour of f(u)/u, the above equation has at least four non-trivial solutions, two of them sign changing.

We consider the existence of stationary or pinned waves of reaction–diffusion equations in heterogeneous media. By combining averaging, homogenization and dynamical-systems techniques we prove under mild non-degeneracy conditions that if the heterogeneity is periodic with period ε, pinned solutions persist at most for intervals in parameter space whose length is O(e−c/√ε).

We investigate geometrically exact generalized continua of micromorphic type in the sense of Eringen. The two-field problem for the macrodeformation φ and the affine microdeformation P̄ ∈ GL+(3, R) in the quasistatic, conservative load case is investigated in a variational form. Depending on material constants, two existence theorems in Sobolev spaces are given for the resulting nonlinear boundary-value problems. These results comprise existence results for the micro-incompressible case P̄ ∈ SL(3, R) and the Cosserat micropolar case P̄ ∈ SO(3, R). In order to treat external loads, a new condition, called bounded external work, has to be included, which overcomes the conditional coercivity of the formulation. The possible lack of coercivity is related to fracture of the micromorphic solid. The mathematical analysis uses an extended Korn first inequality. The methods of choice are the direct methods of the calculus of variations.

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.

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.

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.

The long-time behaviour of solutions to a semilinear damped wave equation in a three-dimensional bounded domain with the nonlinearity rapidly oscillating in time (f = f(ε, u, t/ε)) is studied. It is proved that (under natural assumptions) the behaviour of solutions whose initial energy is not very large can be described in terms of global (uniform) attractors Aε of the corresponding dynamical processes and that, as ε → 0, these attractors tend to the global attractor A0 of the corresponding averaged system. We also give the detailed description of these attractors in the case where the limit attractor A0 is regular.

Moreover, we give explicit examples of semilinear hyperbolic equations where the uniform attractor Âε (for the initial data belonging to the whole energy phase space) contains the irregular resonant part, which tends to infinity as ε → 0, and formulate the additional restrictions on the nonlinearity f which guarantee that this part is absent.

In this paper we consider the Navier–Stokes equations in Rn, n ≥ 3. We prove the asymptotic stability for weak solutions in the marginal class u ∈ L2(0, ∞; BMO), where ‘BMO’ denotes the bounded mean oscillation function, with arbitrary initial and external perturbations.