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This paper is concerned with the following H’enon type problem where B1(0) is the unit ball in ℝN centered at the origin, 1 < p, q and if N ≥ 3. We show that there exists α∗ > 0 such that the ground state solution of the problem is non-radial if α > α∗. We also consider the limiting behavior of the ground state solution (u, v) as p + q → 2∗. We prove that the maximum points of two components u and v concentrate at the same point on the boundary ∂B1(0).

In this paper we study the generalized Hénon equation in the unit ball, where the coefficient function may or may not change its sign. We prove that the least energy solution is not radial and moreover we show the existence of a group invariant positive solution without radial symmetry.

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson system where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution for all λ ≠ 0 and sufficiently small ε > 0. Moreover, for each λ ≠ 0 we show that uε converges to the positive ground state solution of the associated limit problem and concentrates to a maximum point of b(x) in certain sense as ε → 0. Furthermore, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.

In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension $2n+2$ which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on $\partial W$, which we call \emph{convex open book}, induced by a compact convex Lefschetz fibration on $W$ carries the contact structure induced by the convex symplectic structure (i.e., Liouville structure) on $W$. Here we show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on $W$, any simply connected embedded Lagrangian submanifold of a page in a convex open book on $\partial W$ can be assumed to be Legendrian in $\partial W$ with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. Moreover, a result of Akbulut-Arikan implies that there is a one-to-one correspondence between convex stabilizations of a convex open book and convex stabilizations of the corresponding compact convex Lefschetz fibration. We also show that the convex stabilization of a compact convex Lefschetz fibration on $W$ yields a compact convex Lefschetz fibration on a Liouville domain $W'$ which is exact symplectomorphic to a \emph{positive expansion} of $W$. In particular, with the induced structures $\partial W$ and $\partial W'$ are contactomorphic., Comment: 13 pages, 1 figure, minor corrections made

In this paper we study a class of quasilinear problems, in particular we deal with multiple sign-changing solutions of quasilinear elliptic equations. We further develop an approach used in our earlier work by exploring elliptic regularization. The method works well in studying multiplicity and nodal property of solutions.

Let Ω be a bounded domain in ℝN. In this paper, we consider the following nonlinear elliptic equation of N-Laplacian type: when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝN when the nonlinear term f has the exponential growth only deal with the case when f satisfies the (AR) condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami [9, 10, 21]. Examples of f and comparison with earlier assumptions in the literature are given.

The system describes pulses in nonlinear fiber couplers. It has the family (U1+λ, -U1+λ), -1 < λ < ∞, of soliton states (that is, homoclinic solutions to the origin), where For λ ≥ 1, the equilibrium (0, 0) is not hyperbolic and therefore the soliton state (U1+λ, −U1+λ) can be qualified as “singular”. In N. Akhmediev and A. Ankiewicz [1], it is observed numerically that a branch of homoclinic solutions bifurcates subcritically at λ = 1 from the family (U1+λ, −U1+λ). The aim of the present paper is to give a rigorous proof of the existence of this bifurcation, as desired in A. Ambrosetti and D. Arcoya [3]. A particular feature of the present problem is that the linearized system at (U2, −U2) has a non-constant bounded solution that does not vanish at infinity. Hence the bifurcating homoclinic solutions have a transient “spatial” region where they are well described with the help of this bounded function. Moreover the decay to 0 is governed by two different scales, the larger one originating from the singular aspect of (U2, −U2). The existence proof developed here relies on the “broken geodesic” technique to match the inside transient region with the outside region.

In this paper, we shall discuss Bargmann type measures on C for several classes of probability measures on R. The unified interpolation expressions include not only the classical Bargmann measure and its q-deformation, but also their t-deformations and dilations. As a special case, we get conditions on existence and an explicit form of the Bargmann representation for the free Meixner family of probability measures.

In this paper, we introduce semi-slant submersions from almost product Riemannian manifolds onto Riemannian manifolds. We give some examples, investigate the geometry of foliations which are arisen from the definition of a Riemannian submersion. We also find necessary and sufficient conditions for a semi-slant submersion to be totally geodesic.

In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving "big image" results for symplectic Galois representations., Comment: 17 pages. This manuscript is extracted from an old version of our paper arXiv:1203.6552

The device named autoclave used in pharma industries has been studied in this paper. This study provides the discussion how to obtain the reliability testing strategy of two dis-similar parallel units. The system is considered to be in operative condition if at least one out of two is in operative state. A single repair facility is available for repairing both kinds of failed units. In addition to repair mechanism, inspection policy has also been introduced for failed automatic unit. But the manual one does not require any such supervision facilty. Various essential measures of system effectiveness such as mean time to system failure (MTSF), steady state availability, busy period of supervisor and repairman are examined probabilistically by using geometric distribution and regenerative point techniques. A graph has been plotted to represent the behaviour of profit function and MTSF with respect to different failure and repair rate.

In this paper, we characterize local exponential monomials and polynomials on different types of Abelian groups and we prove Montel-type theorems for these function classes.

The purpose of this paper is to study convergence of a newly defined modified S-iteration process to a common fixed point of two asymptotically quasi-nonexpansive type mappings in the setting of CAT(0) space. We give a suffcient condition for convergence to a common fixed point and establish some strong convergence theorems for the said iteration process and mappings under suitable conditions. Our results extend and improve many known results from the existing literature.

In this paper, we study the existence of solutions to boundary blow-up elliptic problems

In this paper, we introduce a concept of piecewise weighted pseudo almost periodic functions on a Banach space and present the essential properties of this kind of functions, including the composition theorems and the uniqueness of decomposition of the functions. As an application, we give some results on the existence and stability of piecewise weighted pseudo almost periodic mild solutions to an abstract impulsive differential equation. Moreover, a concrete example is given to illustrate our abstract results.

In the paper, the authors find two sharp and double inequalities for bounding the second Seiffert mean either by a one-parameter family of means derived from the centroidal mean or by a convex combination of the arithmetic and contra-harmonic means.

In this paper, we mainly investigate a necessary and sufficient condition for a

In this paper, the notion of

Expectation is the fundamental concept in statistics and probability. As two generalizations of expectation, Choquet and Choquet-like expectations are commonly used tools in generalized probability theory. This paper considers the Stolarsky inequality for two classes of Choquet-like integrals. The first class generalizes the Choquet expectation and the second class is an extension of the Sugeno integral. Moreover, a new Minkowski’s inequality without the comonotonicity condition for two classes of Choquet-like integrals is introduced. Our results significantly generalize the previous results in this field. Some examples are given to illustrate the results.

In this paper, we study the existence of multiple solutions of fractional impulsive integrodifferential equations with nonlinear boundary conditions. By applying the Amann theorem and the method of upper and lower solutions, we obtain some new results on the multiple solutions.

In this paper, we consider the concept of the so-called generalized metric space and give some results on the existence, uniqueness and estimation of the solutions of Fredholm type integrodifferential equations in two variables using Perov’s fixed point theorem. Furthermore, we give some illustrative examples to verify the effectiveness and applicability of our main result.

The mod 2 cohomology of the real flag manifolds is known to be isomorphic to a polynomial algebra modulo a certain ideal. In this paper reduced Gröbner bases for these ideals are obtained in the case of manifolds

In this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled by where Ω is a bounded domain of RN, N ≥ 3, with smooth boundary, A is the elliptic operator, not in divergence form, given by Here aij, ai , i,j = 1,...,N , and a0 satisfy suitable regularity conditions, while 1 < s < 4/(N − 2) and the obstacle ψ is a function sufficiently smooth. Even if this problem is not variational in nature, we will prove the existence of non-trivial non-negative solutions for it, performing a variational approach combined with a penalization technique. This kind of approach seems to be new for problems of this type. We also prove a C1,α-regularity result for the solutions of our problem.

In this paper, we apply the discrete Littlewood-Paley-Stein analysis to prove the duality theorem of weighted multi-parameter Hardy spaces associated with Zygmund dilations, i.e., (HpZ (ω))∗ = CMOZ p (ω) for 0 < p ≤ 1. Our theorems extend the HZ p - CMOZ p duality theorems established in [13] (see also [12]) for non-weighted multi-parameter Hardy spaces associated with the Zygmund dilation.

We give a self-contained proof of the A2 conjecture, which claims that the norm of any Calderón–Zygmund operator is bounded by the first degree of the A2 norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condition by the last three authors. Here we replace this reduction by a new weighted norm bound for dyadic shifts – linear in the A2 norm of the weight and quadratic in the complexity of the shift –, which is based on a new quantitative two-weight inequality for the shifts. These sharp one- and two-weight bounds for dyadic shifts are the main new results of this paper. They are obtained by rethinking the corresponding previous results of Lacey–Petermichl–Reguera and Nazarov–Treil–Volberg. To complete the proof of the A2 conjecture, we also provide a simple variant of the representation, already in the original proof, of an arbitrary Calderón–Zygmund operator as an average of random dyadic shifts and random dyadic paraproducts. This method of the representation amounts to the refinement of the techniques from non-homogeneous Harmonic Analysis.

In this paper, we show that the following system −Δu + λV(x)u = g(x, v), −Δv + λV(x)v = f (x, u), x ∈ ℝN (0.1) possesses at least one non-trivial solution pair (u, v) for λ > 0 large enough, where f (x, t), g(x, t) are continuous functions on ℝN × ℝ and super-linear at t = 0 as well as at t = +∞, V(x) is allowed to be sign-changing. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition.

This is a continuation of our previous paper Chang, Wang, Zhang [6]. We investigate the multiple non semi-trivial solutions for nonlinear elliptic systems by two kinds of index theory. In particular the pseudo index theory for the Z2 × Z2 index theory is developed

This paper is devoted to the study of the nonlinear elliptic problem with supercritical critical exponent (Pε) : −Δu = K|u|4/(n−2)+εu in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 3, K is a C3 positive function and ε is a positive real parameter. We show first that in dimension 3, for ε small, (Pε) has no sign-changing solutions with low energy which blow up at two points. For n ≥ 4, we prove that there are no sign-changing solutions which blow up at two nearby points. We also show that (Pε) has no bubble-tower sign-changing solutions.

Let E $E$ be an elliptic curve over ${\mathbb {Q}}$ without complex multiplication. For each prime p $p$ of good reduction, let | E ( p ) | $|E({\mathbb {F}}_p)|$ be the order of the group of points of the reduced curve over p ${\mathbb {F}}_p$ . According to a conjecture of Koblitz, there should be infinitely many such primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ is prime, unless there are some local obstructions predicted by the conjecture. Suppose that E $E$ is a curve without local obstructions (which is the case for most elliptic curves over ${\mathbb {Q}}$ ). We prove in this paper that, under the GRH, there are at least 2 . 778 C E twin x / ( log x ) 2 $2.778 C_E^{\rm twin} x / (\log x)^2$ primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ has at most 8 prime factors, counted with multiplicity. This improves previous results of Steuding & Weng [20, 21] and Miri & Murty [15]. This is also the first result where the dependence on the conjectural constant C E twin $C_E^{\rm twin}$ appearing in Koblitz's conjecture (also called the twin prime conjecture for elliptic curves) is made explicit. This is achieved by sieving a slightly different sequence than the one of [20] and [15]. By sieving the same sequence and using Selberg's linear sieve, we can also improve the constant of Zywina [24] appearing in the upper bound for the number of primes p $p$ such that | E ( p ) | $|E({\mathbb {F}}_p)|$ is prime. Finally, we remark that our results still hold under an hypothesis weaker than the GRH.

In this paper we consider a biharmonic equation of the form Δ2u+V(x)u = f (u) in the whole four-dimensional space ℜ4. Assuming that the potential V satisfies some symmetry conditions and is bounded away from zero and that the nonlinearity f is odd and has subcritical exponential growth (in the sense of an Adams’ type inequality), we prove a multiplicity result. More precisely we prove the existence of infinitely many nonradial sign-changing solutions and infinitely many radial solutions in H2(ℜ4). The main difficulty is the lack of compactness due to the unboundedness of the domain ℜ4 and in this respect the symmetries of the problem play an important role.

The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of [4], the existence of at least three nontrivial solutions to the quasilinear elliptic equation −Δp(x)u = |u|q(x)−2u + λ f (x, u) in a smooth bounded domain Ω of RN with homogeneous Dirichlet boundary conditions on ∂Ω. We assume that {q(x) = p∗(x)} ≠ ø, where p∗(x) = Np(x)/(N − p(x)) is the critical Sobolev exponent for variable exponents and Δp(x)u = div(|∇u|p(x)−2∇u) is the p(x)−laplacian. The proof is based on variational arguments and the extension of concentration compactness method for variable exponent spaces. Fil: Silva, Analia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina

In this paper, we give a weighted version of regularity of solutions of the Wente problem associated to the modified Helmholtz operator -Δ + αI, where α is a positive constant.

In this paper we consider the following nonlinear elliptic equation with Dirichlet boundary conditions: -Δu = K(x)up, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a smooth domain in ℝn, n ≥ 4 and is the critical Sobolev exponent. Using dynamical and topological methods involving the study of critical points at infinity we establish, under generic conditions on K, some existence and multiplicity results.

We find coexistence states for systems in three dimensions with cyclic dynamic in the boundary of ℝ+ 3. This problem had been already studied for competitive systems. In this paper we consider any interaction. This extension seems to be meaningful since the cyclic dynamic on the boundary can appear in natural way in many interactions.

The purpose of the paper is to establish the existence of ground state solutions to -Δu = ηa(x)f(u)+γb(x)g(u), where the locally Hölder continuous non-negative functions a and b and the non-linearities f and g satisfy some general conditions.

In this paper, we study the effect of domain shape on the existence of 2-nodal solutions for a semilinear elliptic equation involving non-odd nonlinearities.

Earlier papers have studied the existence of solutions of reversible Hamiltonian systems that are heteroclinic between a pair of minimizing periodic solutions. This note extends some of that work by allowing more degenerate behavior for the class of periodic minimizers.

We study the quasilinear Schrödinger equation izt = −∆z +W(x)z − η(|z|2)z − k[∆p(|z|2)]p′(|z|2)z in ℝ2, where W : ℝ2 →ℝ is a positive potential and the nonlinearity η : ℝ2 × ℝ → ℝ has critical or sub-critical exponential growth. Quasilinear Schrödinger equations of this type have been studied as models of several physical phenomena such as superfluid film equation, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics and in condensed matter theory. In a suitable Orlicz space together with Trudinger-Moser inequality we establish an existence of standing wave solutions for this problem. The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics.

In this paper we consider nonlinear Dirichlet problems driven by the p-Laplacian differential operator with a nonsmooth potential (hemivariational inequalities). We assume that the problem is resonant at infinity with respect to λ1 > 0 (the principal eigenvalue of the Dirichlet p-Lapalcian) from the right. Using minimax methods based on the nonsmooth critical point theory we prove an existence and a multiplicity theorem.

In this paper, we consider a singular elliptic system with both concave-convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

In this paper we consider existence, asymptotic behavior near the boundary and uniqueness for solutions to Δu = eq(x)u in a bounded smooth domain Ω with the boundary condition u(x) → + ∞ as dist(x, ∂Ω) → 0. The exponent q(x) is assumed to be a Hölder continuous function which is either positive on ∂Ω or is positive in a neighborhood of ∂Ω maybe vanishing on ∂Ω. When dealing with nonnegative exponents q we are allowing nonempty interior regions Ω0 ⊂ Ω where q vanishes. Changing sign exponents q will be also considered.

We are interested in the existence of standing waves for the nonlinear Klein Gordon equation ε2□ψ + Wʹ(ψ) = 0 in a bounded domain D. A standing wave has the form ψ(t, x) = u(x)e-iwt/ε; for these solutions the Klein Gordon equation becomes We want to use a Benci-Cerami type argument in order to prove a the existence of several standing waves localized in suitable points of D. The main result of this paper is that, under suitable growth condition on W, for ε suffciently small, we have at least cat(D) stationary solutions of equation (†), where cat(D) is the Ljusternik-Schnirelmann category. The proof is achieved by solving a constrained critical point problem via variational techniques.

This paper is concerned with the following nonlinear Schrödinger equation: where f(x, t)t-1 < μ* := inf σ(-Δ + V) < f(x, t)t-1 < V(∞) = V(x). The existence of a positive solution and a least energy solution for problem (P) are discussed. Here, we assume neither that f(x, t) ≡ f(t), nor that f(x, t)t-1 is non-decreasing on t ≥ 0. Furthermore, we do not require that f(x, t) have a limit as |x| → +∞.

In this paper we will establish two different classes of ellipticity criteria, called the Lp criteria and the Lp-Lq criteria respectively, for planar linear Hamiltonian systems with periodic coefficients. The criteria are explicitly expressed using the Lp and Lq norms of coefficients and some known Sobolev constants. These results can be considered as the extensions of the famous Lyapunov stability criterion for Hill’s equations.

In this paper, we are interested in the following biharmonic equation: with Navier boundary conditions on ∂Ω; where Ω is an open bounded domain of ℝn, with n ≥ 5, having a smooth boundary, and ∑ ⊂ Ω is either a set of finite number of points of Ω or a smooth closed submanifold of Ω without boundary. We construct positive solutions to this problem which are singular at ∑.

The well-known Belousov-Zhabotinskii system can be written as ẋ = s(x + y - qx2 - xy), ẏ = s-1(-y + fz - xy), ż = w(x - z) with f, q, s, w ∈ ℝ and s ≠ 0. In this paper we characterize its global analytic first integrals.

This paper deals with the instability of the ground state solitary wave solution to the Klein-Gordon-Zakharov system in three space dimensions with c ≥ 1, which is a model to describe the Langmuir turbulence in plasma. First we construct a suitable constrained variational problem and obtain the existence of the standing waves with ground state by using variational calculus and scaling argument. Then by defining invariant sets and applying some priori estimates, we prove the orbital instability of the ground state in the following sense: in each neighborhood of it, there exists a solution whose energy diverges in finite or infinite time.

Abelian gauge theories consist of a class of field equations which provide a model for the interaction between matter and electromagnetic fields. In this paper we analyze the existence of solitary waves for these theories. We assume that the lower order term W is positive and we prove the existence of solitary waves if the coupling between matter and electromagnetic field is small. We point out that the positiveness assumption on W implies that the energy is positive: this fact makes these theories more suitable to model physical phenomena., 27 pages

Let (𝑋, 𝑑) be a convex metric space, 𝐶 be a closed and convex subset of 𝑋 and let 𝐵(𝐶) be the family of all nonempty bounded subsets of 𝐶. In this paper some results are obtained on the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings 𝑆,𝑇 : 𝐶 → 𝐵(𝐶) which satisfy condition (2.1) below.

This paper is concerned with the existence of positive solutions for a class of quasilinear elliptic equations on ℝ. The results are proved by combining the concentration-compactness principle due to Lions with a minimization approach.