In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.

We consider the fractional Schrödinger equations with focusing Hartree-type nonlinearities. When the energy is negative, we show that the solution blows up in a finite time. For this purpose, based on Glassey’s argument, we obtain a virial-type inequality.

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely, the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbour site percolation cluster at a non-trivial percolation threshold.

We show that non-occurrence of the Lavrentiev phenomenon does not imply that the singular set is small. Precisely, given a compact Lebesgue null subset E ⊆ ℝ and an arbitrary superlinearity, there exists a smooth strictly convex Lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactly E but still admit approximation in energy by smooth functions.

In dynamo theory the distinction between decaying (in time) magnetic fields and those that do not is of crucial importance. Often decay is not manifest for the magnetic field itself but only for a single component or a scalar potential. Typically these auxiliary quantities satisfy evolution equations of the same type as the original induction equation. We prove here for these equations a theorem relating the decay of a solution to the decay of its higher derivatives. This result allows us to relate the decay of an auxiliary quantity to that of the magnetic field and, moreover, to relate integral decay to pointwise decay. As an application we strengthen the ‘toroidal velocity theorem’ in that we demonstrate pointwise decay of the magnetic field and electric current to zero under the conditions of this theorem.

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

In 1994 Boyd derived a resurgence representation for the gamma function, exploiting the 1991 reformulation of the method of steepest descents by Berry and Howls. Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities and asymptotics for the late coefficients. The main aim of this paper is to modify Boyd’s resurgence formula, making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.

We prove monotonicity properties of positive weak solutions to – Δpu = f(u) in half-spaces for the case in which (2N + 2)/(N + 2) < p < 2 when zero Dirichlet boundary conditions are imposed. The model nonlinearity is given by f(s) := sq − λsm with q > m ≥ p − 1 and λ > 0.

In this paper, we study the zero dissipation limit of the initial boundary-value problem of the multi-dimensional Boussinesq equations with viscosity and heat conductivity. Such equations are used as models for the motion of multi-dimensional incompressible fluids in atmospheric and oceanographic turbulence. In particular, they describe the thermal convection of an incompressible flow, and constitute the relations between the velocity field, the pressure and the local temperature. Under the Navier slip boundary condition in the velocity field and the thermal isolation boundary condition for the temperature, we prove the existence of weak amplitude characteristic boundary layers. Then, by a standard energy method, we prove the L2 convergence of the solutions when both the viscosity and the heat conductivity coefficients tend to 0.

We show that a differential module is Gorenstein projective (injective, respectively) if and only if its underlying module is Gorenstein projective (injective, respectively). We then relate the Ringel–Zhang theorem on differential modules to the Avramov–Buchweitz–Iyengar notion of projective class of differential modules and prove that for a ring R there is a bijective correspondence between projectively stable objects of split differential modules of projective class not more than 1 and R-modules of projective dimension not more than 1, and this is given by the homology functor H and stable syzygy functor ΩD. The correspondence sends indecomposable objects to indecomposable objects. In particular, we obtain that for a hereditary ring R there is a bijective correspondence between objects of the projectively stable category of Gorenstein projective differential modules and the category of all R-modules given by the homology functor and the stable syzygy functor. This gives an extended version of the Ringel–Zhang theorem.

In this paper we study the problem concerning stability and asymptotic behaviours of solutions for a degenerate Landau–Lifshitz equation in micromagnetics involving only the non-local magnetostatic energy. Due to the lack of derivative estimates, we do not have the compactness needed for strong convergence and the natural convergence is weak* convergence. By formulating the problem in a new framework of differential inclusions, we show that the Cauchy problems for such an equation are not stable under the weak* convergence of initial data. For the asymptotic behaviours of weak solutions, we establish an estimate on the weak* ω-limit sets that is valid for all initial data satisfying the saturation condition.