In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or nonexistence properties, power- and exponential-type Sobolev regularity results, and the boundary behaviour of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.

We consider the following equations involving negative exponent:

where p > 0. Under optimal conditions on the parameters α > −2 and p > 0, we prove the non-existence of finite Morse index solution on exterior domains or near the origin. We also prove an optimal regularity result for solutions with finite Morse index and isolated rupture at 0.

It is a well-known problem to find for n letters the number of the arrangements in which no letter occupies its original place; and the solution of it is given by the following general theorem :— viz., the number of the arrangements which satisfy any r conditions is

where 1 denotes the whole number of arrangements; (1) the number of them which fail in regard to the first condition; (2) the number which fail in regard to the second condition; (12) the number which fail in regard to the first condition, and also in regard to the second condition; and so on: Σ(1) means (1) + (2) … + (r) : Σ(12) means (12) + (13) + (2r) … + (r − 1, r); and so on, up to (12…r), which denotes the number failing in regard to each of the r conditions.

We consider weak solutions to

with p > 1, q ≥ max{p − 1, 1}. We exploit the Moser iteration technique to prove a Harnack comparison inequality for C1 weak solutions. As a consequence we deduce a strong comparison principle.

In the Inverse Spectral Theorem in the form given by Levitan and Gasymov, necessary and sufficient conditions are given for a non-decreasing function, p(ℷ), to be the spectral function of a Sturm–Liouville problem. In these conditions, p(ℷ) is compared with the spectral function for the particular Strum–Liouville problem

If the method of Levitan and Gasymov's proof is slightly adapted, the necessary and sufficient conditions can be stated in a more general form in which p(ℷ) is compared with the spectral function for any problem of the form

where hO is real and qo(x) locally integrable.

Though in my former papers on knots I have made but little allusion to cases in which two or more closed curves are linked together, the method I have employed is easily and directly applicable to them. I stated to the British Association that the number of intersections passed through in going continuously along a curve, from any intersection to the same again, is always even—whether it be linked with other curves or not Hence, even when a number of closed curves are linked together, the intersections may be so arranged as to be alternately over and under along each of the curves.

Conditions are obtained under which radial solutions of generalised capillary-type equations exhibit a first vertical point, together with bounds and asymptotic estimates for these points.

We show that the linear viscous damping Δut, is so strong that it altogether prevents propagation of singularities of the gradient of solutions to the system of viscoelasticity. Moreover, no creation or annihilation of singularities is possible in finite time.

In this paper, we are concerned with the existence of unbounded orbits of the mapping

\begin{align*} \theta_1&=\theta+2\pi+\frac{1}{\rho}\mu(\theta)+o(\rho^{-1}), \\ \rho_1&=\rho+c-\mu'(\theta)+o(1),\quad\rho\to\infty, \end{align*}

where $c$ is a constant and $\mu(\theta)$ is $2\pi$-periodic. Assume that $c\not=0$, that $\mu(\theta)$ is non-negative (or non-positive) and that $\mu(\theta)$ has finitely many degenerate zeros in $[0,2\pi]$. We prove that every orbit of the given mapping tends to infinity in the future or in the past for sufficiently large $\rho$. On the basis of this conclusion, we further prove that the equation $x''+f(x)x'+V'(x)+\phi(x)=p(t)$ has unbounded solutions provided that $V$ is an isochronous potential at resonance and $F(x)$ ($F(x)=\int_0^xf(s)\,\mathrm{d} s$) and $\phi(x)$ satisfy some limit conditions. Meanwhile, we also obtain the existence of $2\pi$-periodic solutions of this equation.

Let X be a set with infinite cardinality m and let B be the Baer-Levi semigroup, consisting of all one-one mappings a:X→X for which ∣X/Xα∣ = m. Let Km=<B 1B>, the inverse subsemigroup of the symmetric inverse semigroup ℐ(X) generated by all products β−γ, with β,γ∈B. Then Km = <N2>, where N2 is the subset of ℐ(X) consisting of all nilpotent elements of index 2. Moreover, Km has 2-nilpotent-depth 3, in the sense that

Let Pm be the ideal {α∈Km: ∣dom α∣<m} in Km and let Lm be the Rees quotient Km/Pm. Then Lm is a 0-bisimple, 2-nilpotent-generated inverse semigroup with 2-nilpotent-depth 3. The minimum non-trivial homomorphic image of Lm also has these properties and is congruence-free.

We show that lattice dynamical systems naturally arise on infinite-dimensional invariant manifolds of reaction–diffusion equations with spatially periodic diffusive fluxes. The result connects wave-pinning phenomena in lattice differential equations and in reaction–diffusion equations in inhomogeneous media. The proof is based on a careful singular perturbation analysis of the linear part, where the infinite-dimensional manifold corresponds to an infinite-dimensional centre eigenspace.

We study a class of Schrödinger lattice systems with sublinear nonlinearities and perturbed terms. We get an interesting result that the systems do not have nontrivial homoclinic solutions if the perturbed terms are removed, but the systems have ground state homoclinic solutions if the perturbed terms are added. Besides, we also study the continuity of the homoclinic solutions in the perturbation terms at zero. To the best of our knowledge, there is no published result focusing on the perturbed Schrödinger lattice systems.

General convergence theorems are established for the lower part of the intermediate operator spectra arising in lower bound methods of Aronszajn type.Convergence of the T*T method is considered in detail, as are constructions that allow perturbations that are nonclosable in the underlying Hilbert space. Problems with essential spectra are admitted, as is the use of infinite rank projections capable of displacing essential spectra. These last features are necessary for a theory applicable to Schrodinger operators corresponding to atomic systems.

We study the effect of the potential |y|α on the stability of entire solutions for elliptic equations on ℝN, N ≥ 2, with exponential or smoooth/singular polynomial nonlinearities. Instability properties are crucial in order to establish regularity of the extremal solution to some related Dirichlet nonlinear eigenvalue problem on bounded domains. As a by-product of our results, we will improve the known results about the regularity of such solutions.

Let H be a separable Hilbert space and let CL(H) be the semigroup of continuous, linear maps from H to H. Let E+ be the idempotents of CL(H). Let Ker ɑ and Im ɑ be the null-space and range, respectively, of an element ɑ of CL(H) and let St ɑ be the subspace {x∊H: xɑ = x} of H. It is shown that 〈E+〉 = I∪F∪{i}, where

and ι is the identity map. From the proof it is clear that I and F both form subsemigroups of 〈E+〉 and that the depth of I is 3. It is also shown that the depths of F and 〈E+〉 are infinite.

This paper uses previous results of Chillingworth, Marsden and Wan on symmetry and bifurcation for the traction problem in three dimensional elastostatics to establish new results on the Signorini expansion. We show that the Signorini compatibility conditions are necessary and sufficient for linearization stability and analogies with results known for other field theories are pointed out. Under an explicit non-degeneracy condition, a new series expansion is given in which successive terms are inductively determined in pairs rather than singly. Our results include as special cases, classical results of Signorini, Tolotti and Stoppelli.

We propose and investigate a stage-structured SLIRM epidemic model with latent period in a spatially continuous habitat. We first show the existence of semi-travelling waves that connect the unstable disease-free equilibrium as the wave coordinate goes to − ∞, provided that the basic reproduction number $\mathcal {R}_0 > 1$ and $c > c_*$ for some positive number $c_*$. We then use a combination of asymptotic estimates, Laplace transform and Cauchy's integral theorem to show the persistence of semi-travelling waves. Based on the persistent property, we construct a Lyapunov functional to prove the convergence of the semi-travelling wave to an endemic (positive) equilibrium as the wave coordinate goes to + ∞. In addition, by the Laplace transform technique, the non-existence of bounded semi-travelling wave is also proved when $\mathcal {R}_0 > 1$ and $0 < c < c_*$. This indicates that $c_*$ is indeed the minimum wave speed. Finally simulations are given to illustrate the evolution of profiles.

This paper is concerned with the dynamics of nonlinear one-dimensional beam equations. We consider nonlinear beam equations with viscosity or with a lower-order damping term instead of the viscosity, and we establish the existence of global attractors for both systems.

By giving some new treatments we can improve a classical result of Walfisz (1963) on the asymptotic formula of Euler's function.

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.