The existence of a non-trivial bounded solution to the Dirichlet problem is established for a class of nonlinear elliptic equations involving a fully anisotropic partial differential operator. The relevant operator depends on the gradient of the unknown through the differential of a general convex function. This function need not be radial, nor have a polynomial-type growth. Besides providing genuinely new conclusions, our result recovers and embraces, in a unified framework, several contributions in the existing literature, and augments them in various special instances.

We develop various facets of the theory of quadratic forms on a Hilbert space suggested by a criterion of Kato which characterizes closed forms in terms of lower semicontinuity.

This paper deals with the influence of the Hardy potential in a semilinear heat equation. Precisely, we consider the problem

where Ω⊂ℝN, N≥3, is a bounded regular domain such that 0∈Ω, p>1, and u0≥0, f≥0 are in a suitable class of functions.

There is a great difference between this result and the heat equation (λ=0); indeed, if λ>0, there exists a critical exponent p+(λ) such that for p≥p+(λ) there is no solution for any non-trivial initial datum.

The Cauchy problem, Ω=ℝN, is also analysed for 1<p<+(λ). We find the same phenomenon about the critical power p+(λ) as above. Moreover, there exists a Fujita-type exponent, F(λ), in the sense that, independently of the initial datum, for 1<p<F(λ), any solution blows up in a finite time. Moreover, F(λ)>1+2/N, which is the Fujita exponent for the heat equation (λ=0).

Solutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.

In this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.

The symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.

We characterize injective continuous maps on the space of real or complex rectangular matrices preserving adjacent pairs of matrices. We also extend Hua's fundamental theorem of the geometry of rectangular matrices to the infinite-dimensional case. An application in the theory of local automorphisms is presented.

An element α of Pn, the semigroup of all partial transformations of {1,2,…, n}, is said to have projection characteristic (r, s), or to belong to the set [r, s], if dom α= r, im α = s. Let E be the set of all idempotents in Pn\[n, n] and E1, the set of those idempotents with projection characteristic (n, n − 1) or (n − 1, n − 1). For α in Pn\[n, n], we define a number g(α), called the gravity of α and closely related to the number denned in Howie [5] for full transformations, and we obtain the result that

Let d(α) be the defect of α, and for any real number x let [x] be the least integer m such that m ≧ x. Then by analogy with the results of Saito [9] we have that

α ϵ Ek(α) and α ∉ Ek(α)

where k(α) = [g(α)/d(α)] or [g(α)/d(α)+ 1. Following Howie, Lusk and McFadden [6] we then explore connections between the defect and the gravity of α. Letting K(n, r) be the subsemigroup of Pn consisting of all elements of rank r or less, we prove a result, corresponding to that of Howie and McFadden [7] for total transformations, that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n + 1, r + 1), the Stirling number of the second kind.

The entire positive solutions of a conformally invariant biharmonic equation in Rn will be classified using the method of moving spheres. As a byproduct, one also shows that any entire non-negative solution of the equation Δ2u = up with 1 ≤ p < (n + 4)/(n−4) with n ≥ 2 is zero.

We prove that critical growth problems for polyharmonic operators admit nontrivial solutions for a wide class of lower-order perturbations of the critical term. The results highlight the phenomenon of bifurcation of the critical dimensions discovered by Pucci and Serrin; moreover, we show that another bifurcation seems to appear for ‘nonresonant’ dimensions.

The Kadomtsev–Petviashvili (KP) equation can be formally derived as an envelope equation for three-dimensional unidirectional water waves in the limit of long waves. As a first step towards a mathematical justification, we consider here a two-dimensional Boussinesq equation, which is a realistic model for three-dimensional water waves. Using rigorous estimates, we show that part of the dynamics of the KP equation can be found approximately in the two-dimensional Boussinesq equation. On the other hand, there exist initial data for the KP equation such that the corresponding solutions of the two-dimensional Boussinesq equation behave in no way according to the KP prediction. We expect that similar results hold for the three-dimensional water wave problem too.

In this paper, we study the controllability of second-order nonlinear stochastic delay systems driven by the Rosenblatt distributions in finite dimensional spaces. A set of sufficient conditions are established for controllability of nonlinear stochastic delay systems using fixed point theory, delayed sine and cosine matrices and delayed Grammian matrices. Furthermore, controllability results for second-order stochastic delay systems driven by Rosenblatt distributions via the representation of solution by delayed sine and cosine functions are presented. Finally, our theoretical results are illustrated through numerical simulation.

Let n ≥ 3, Ω ⊂ Rn be a domain with 0 ∈ Ω, then, for all the Hardy–Sobolev inequality says that and equality holds if and only if u = 0 and ((n − 2)/2)2 is the best constant which is never achieved. In view of this, there is scope for improving this inequality further. In this paper we have investigated this problem by using the fundamental solutions and have obtained the optimal estimates. Furthermore, we have shown that this technique is used to obtain the Hardy–Sobolev type inequalities on manifolds and also on the Heisenberg group.

We consider some geometric aspects of regular Sturm—Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm—Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for separated boundary conditions and those for coupled boundary conditions, or the eigenvalues for self-adjoint boundary conditions and those for non-self-adjoint boundary conditions, are closely related under this structure. Then we give complete characterizations of several subsets of boundary conditions such as the set of self-adjoint boundary conditions that have a given real number as an eigenvalue, and determine their shapes. The shapes are shown to be independent of the differential equation in question. Moreover, we investigate the differentiability of continuous eigenvalue branches under this structure, and discuss the relationships between the algebraic and geometric multiplicities of an eigenvalue.

The main result of this paper is the determination of all plane polynomial vector fields that admit a prescribed collection of algebraic curves as invariant sets. As an application, the polynomial vector fields admitting certain types of algebraic integrating factors are characterized.

Numerous combinatory problems arise in connection with a set of elements subject to a non-associative process of composition—let us say of multiplication—commutative or non-commutative.

Non-associative products may be classified according to their shape. By the shape of a product I mean the manner of association of its factors without regard to their identity. Shapes will be called commutative or non-commutative according to the type of multiplication under consideration. Thus if multiplication is non-commutative, the products (AB.C)D and (BA.C)D are distinct but have the same shape, while D(AB.C) has a different shape. The three expressions, however, have the same commutative shape. I confine attention to products (like these) in which the factors are combined only two at a time.

The rise and decline of epidemics of infectious diseases have been subjects of interest since the earliest times, but the scientific determination of the laws which govern their course offers even yet a wide and almost unworked field. Not but what a large amount of observation has been made on many of the conditions under which epidemics appear and pass away. Many epidemics are seasonal, and these have been studied; but the lack of any means of determining the course which a given epidemic might have taken in the presence of somewhat different conditions has made the deduction of certain conclusions impossible. Even the laws which regulate solitary outbursts of disease, the special subject of this paper, have been little studied. Explanations offered have varied with the period in history.

We consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.

Weak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.

We study the spectrum of regular and singular Sturm–Liouville problems with real-valued coefficients and a weight function that changes sign. The self-adjoint boundary conditions may be regular or singular, separated or coupled. Sufficient conditions are found for (i) the spectrum to be real and unbounded below as well as above and (ii) the essential spectrum to be empty. Also found is an upper bound for the number of non-real eigenvalues. These results are achieved by studying the interplay between the indefinite problems (with weight function which changes sign) and the corresponding definite problems. Our approach relies heavily on operator theory of Krein space.