Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces

The present paper deals with the study of semilinear and non-homogeneous Schrödinger equations on a manifold with conical singularity. We provide a suitable constant by Sobolev embedding constant and for p ∈ (2, 2?) with respect to non-homogeneous term g(x) ∈ L ₂ ^n/2 (B), which helps to find multiple solutions of our problem. More precisely, we prove the existence of two solutions to the problem 1.1 with negative and positive energy in cone Sobolev space H _2,0 ^1,n/2 (B). Finally, we consider p = 2 and we prove the existence and uniqueness of Fuchsian-Poisson problem.     

Global existence and energy decay for a coupled system of Kirchhoff type equations with damping and source terms

This paper deals with the global existence and energy decay of solutions to some coupled system of Kirchhoff type equations with nonlinear dissipative and source terms in a bounded domain. We obtain the global existence by defining the stable set in H ₀ ¹ (Ω) × H ₀ ¹ (Ω), and the energy decay of global solutions is given by applying a lemma of V. Komornik.     

Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

Normal functions and a class of associated boundary functions

Letμ′ be the family of non-empty closed subsets of the Riemann sphere and Λ the family of continuous curves λ with values in the unit disk and lim _t→1 |λ(t)|=1. A meromorphic functionf in |z|<1 induces a mapping\(\hat f\) from Λ intoμ′ by setting\(\hat f\left( \lambda \right)\) equal to the cluster set off on λ. The authors show that if\(\hat f\) is continuous then existence of an asymptotic value ate ^iθ implies the existence of an angular limit. Further if the spherical derivative off iso(1/(1?|z|)) then\(\hat f\) is constant on every open disk in the space Λ.     

On EA-equivalence of certain permutations to power mappings

In this paper we investigate the existence of permutation polynomials of the form x ^d  + L(x) on \({{\mathbb{F}_{2^n}}}\) , where \({{L(x)\in\mathbb{F}_{2^n}[x]}}\) is a linearized polynomial. It is shown that for some special d with gcd(d, 2 ⁿ ?1) > 1, x ^d  + L(x) is nerve a permutation on \({{\mathbb{F}_{2^n}}}\) for any linearized polynomial \({{L(x)\in\mathbb{F}_{2^n}[x]}}\) . For the Gold functions \({{x^{2^i+1}}}\) , it is shown that \({{x^{2^i+1}+L(x)}}\) is a permutation on \({{\mathbb{F}_{2^n}}}\) if and only if n is odd and \({{L(x)=\alpha^{2^i}x+\alpha x^{2^i}}}\) for some \({{\alpha\in\mathbb{F}_{2^n}^{*}}}\) . We also disprove a conjecture in (Macchetti Addendum to on the generalized linear equivalence of functions over finite fields. Cryptology ePrint Archive, Report2004/347, 2004) in a very simple way. At last some interesting results concerning permutation polynomials of the form x ^?1 + L(x) are given.     

An Obstacle Problem for Nonlocal Equations in Perforated Domains

In this paper we analyze the behavior of solutions to a nonlocal equation of the form J ? u (x) ? u (x) = f (x) in a perforated domain Ω ? A ^?? with u = 0 in \(A^{\epsilon } \cup {\Omega }^{c}\) and an obstacle constraint, u ≥ ψ in Ω ? A ^?? . We show that, assuming that the characteristic function of the domain Ω ? A ^?? verifies \(\chi _{\epsilon } \rightharpoonup \mathcal {X}\) weakly ^? in \(L^{\infty }({\Omega })\), there exists a weak limit of the solutions u ^?? and we find the limit problem that is satisfied in the limit. When \(\mathcal {X} \not \equiv 1\) in this limit problem an extra term appears in the equation as well as a modification of the obstacle constraint inside the domain.     

Cylindrical Dyck paths and the Mazorchuk–Turowska equation

We classify all solutions (p, q) to the equation \(p(u)q(u)=p(u+\beta )q(u+\alpha )\) where p and q are complex polynomials in one indeterminate u, and \(\alpha \) and \(\beta \) are fixed but arbitrary complex numbers. This equation is a special case of a system of equations which ensures that certain algebras defined by generators and relations are non-trivial. We first give a necessary condition for the existence of non-trivial solutions to the equation. Then, under this condition, we use combinatorics of generalized Dyck paths to describe all solutions and a canonical way to factor each solution into a product of irreducible solutions.     

On derivation of Euler–Lagrange equations for incompressible energy-minimizers

We prove that any distribution q satisfying the grad-div system \({\nabla q={\rm div}\,{\bf f}}\) for some tensor \({{\bf f}=(f^i_j), \,f^i_j\in h^r(U)\,(1\leq r < \infty}\)) -the local Hardy space; q is in h ^r and q is locally represented by the sum of singular integrals of \({f^i_j}\) with Calderón-Zygmund kernel. As a consequence, we prove the existence and the local representation of the hydrostatic pressure p (modulo constant) associated with incompressible elastic energy-minimizing deformation u satisfying \({|\nabla{\bf u}|^2,\,|{\rm cof}\,\nabla{\bf u}|^2\in h^1}\). We also derive the system of Euler–Lagrange equations for volume preserving local minimizers u that are in the space \({K^{1,3}_{\rm loc}}\) [defined in (1.2)]—partially resolving a long standing problem. In two dimensions we prove partial C ^1,α regularity of weak solutions provided their gradient is in L ³ and p is Hölder continuous.     

Fractional Abstract Cauchy Problems

This paper is concerned with fractional abstract Cauchy problems with order \({\alpha\in(1,2)}\). The notion of fractional solution operator is introduced, its some properties are obtained. A generation theorem for exponentially bounded fractional solution operators is given. It is proved that the homogeneous fractional Cauchy problem (FACP ₀) is well-posed if and only if its coefficient operator A generates an α-order fractional solution operator. Sufficient conditions are given to guarantee the existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem (FACP _f ).     

On the Monge–Ampère equation with boundary blow-up: existence,uniqueness and asymptotics

We consider the Monge–Ampère equation det D ² u = b(x)f(u) > 0 in Ω, subject to the singular boundary condition u = ∞ on ?Ω. We assume that \(b\in C^\infty(\overline{\Omega})\) is positive in Ω and non-negative on ?Ω. Under suitable conditions on f, we establish the existence of positive strictly convex solutions if Ω is a smooth strictly convex, bounded domain in \({\mathbb R}^N\) with N ≥ 2. We give asymptotic estimates of the behaviour of such solutions near ?Ω and a uniqueness result when the variation of f at ∞ is regular of index q greater than N (that is, \(\lim_{u\to \infty} f(\lambda u)/f(u)=\lambda^q\) , for every λ > 0). Using regular variation theory, we treat both cases: b > 0 on ?Ω and \(b\equiv 0\) on ?Ω.     

The rate of decay of stable periodic solutions for Duffing equation with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conditions

For a new class of g(t, x), the existence, uniqueness and stability of \({2\pi}\)-periodic solution of Duffing equation \({x'' + cx' + g(t, x) = h(t)}\) are presented. Moreover, the unique \({2\pi}\)-periodic solution is (exponentially asymptotically stable) and its rate of exponential decay c/2 is sharp. The new criterion characterizes \({g_{x}^{\prime}(t, x) - c^2/4}\) with L ^p -norms \({(p \in [1, \infty])}\), and the classical criterion employs the \({L^{\infty}}\)-norm. The advantage is that we can deal with the case that \({g_{x}^{\prime}(t, x) - c^2/4}\) is beyond the optimal bounds of the \({L^{\infty}}\)-norm, because of the difference between the L ^p -norm and the \({L^{\infty}}\)-norm.     

Fujita type conditions to heat equation with variable source

This paper studies heat equation with variable exponent u _t = Δu + u^p(x) + u ^q in ? ^N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p_? = inf p(x) ≤ p(x) ≤ sup p(x) = p₊. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p₊, q} ≤ 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q ≤ 1 with p₊ > 1, or 1 < q < 1 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p_? ≤ p₊ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p_? > 1 + \(\frac{2}{N}\); and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p_? < 1 + \(\frac{2}{N}\) < p₊.     

On Elliptic Extensions in the Disk

Given two arbitrary functions f ⁽⁰⁾, f ⁽¹⁾ on the boundary of the unit disk D in \({\mathbb R}^2\), it is shown that there exists a second order uniformly elliptic operator L and a function v in L ^p , with L ^p second derivatives (1?p?Lv?=?0 a.e. in D and with v?=?f ⁽⁰⁾ and \(\frac{ \partial v}{\partial n} = f^{(1)}\) on \(\partial{D}\). A similar extension property was proved in Cavazzoni (2003) for any pair of functions f ⁽⁰⁾, f ⁽¹⁾ that are analytic; a result is obtained under weaker regularity assumptions, e.g. with \(\frac{\partial f^{(0)}}{\partial \theta}\) and f ⁽¹⁾ Hölder continuous with exponent \(\eta > \frac{1}{2}\).     

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

We study Darboux-type transformations associated with the focusing nonlinear Schrödinger equation (NLS_) and their effect on spectral properties of the underlying Lax operator. The latter is a formallyJ (but nonself-adjoint) Diract-type differential expression of the form

$M(q) = i\left( {\begin{array}{*{20}c} {\frac{d}{{dx}}} &; { - q} \\ { - \bar q} &; { - \frac{d}{{dx}}} \\ \end{array} } \right)$

(1)

satisfying\({\mathcal{J}} M(q)\mathcal{J} = M(q)^* \), whereJ is defined byJ C, andC denotes the antilinear conjugation map in ?²,\({\mathcal{C}}(a,b)^{\rm T} = (\bar a,\bar b)^{\rm T} ,a,b \in \) ?. As one of our principla results, we prove that under the most general hypothesisq ∈ _loc ¹ (?) onq, the maximally defined operatorD(q) generated byM(q) is actually {itJ}-self-adjoint in inL ²(?)². Moreover, we establish the existence of Weyl-Titchmarsh-type solutions ψ+(z, ·) ?L ² ([R, ∞))² and ψ?(z, ·) ∈L ² ((?∞,R]) for allR∈? ofM(q)Ψ ± (z)=zΨ ± (z) forz in the resolvent set ofD(q).

The Darboux transformations considered in this paper are the analogue of the double commutation procedure familiar in the KdV and Schrödinger operator contexts. As in the corresponding case of Schrödinger operators, the Darboux transformations in question guarantee that the resulting potentialsq are locally nonsingular. Moreover, we prove that the construction ofN-soliton NLS_potentialsq ^(N) with respect to a general NLS background potentialq ?L _loc ¹ (?), associated with the Dirac-type operatorsD(q ^(N) ) andD(q), respectively, amounts to the insertio ofN complex conjugate pairs ofL ²({?}²-eigenvalues\(\{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} \) into the spectrum σ(D(q)) ofD(q), leaving the rest of the spectrum (especially, the essential spectrum σ_e(itD)(q))) invariant, that is,

$\sigma (D(q^{(N)} )) = \sigma (D(q)) \cup \{ z_1 ,\bar z_1 ,...,z_N ,\bar z_N \} ,$

(1)

$\sigma _e (D(q^{(N)} )) = \sigma _e (D(q))$

(1)

These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operatorD(q) and the Dirac operatorD(q ^(N) ) obtained afterN Darboux-type transformations.     

No decreasing sequence of cardinals

In set theory without the Axiom of Choice (AC), we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f (n + 1) ? f (n), where for sets x and y, x ? y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies AC. In this paper, among other results, we show that NDS is a strong axiom by establishing that AC^LO (AC restricted to linearly ordered sets of non-empty sets, and also equivalent to AC in ZF, the Zermelo–Fraenkel set theory minus AC) ? NDS in ZFA set theory (ZF with the Axiom of Extensionality weakened in order to allow the existence of atoms). The latter result provides a strongly negative answer to the question of whether “every Dedekind-finite set is finite” implies NDS addressed in G. H. Moore “Zermelo’s Axiom of Choice. Its Origins, Development, and Influence” and in P. Howard–J. E. Rubin “Consequences of the Axiom of Choice”. We also prove that AC^WO (AC restricted to well-ordered sets of non-empty sets) ? NDS in ZF (hence, “every Dedekind-finite set is finite” ? NDS in ZF, either) and that “for all infinite cardinals m, m + m = m” ? NDS in ZFA.     

Bending the helicoid

We construct Colding–Minicozzi limit minimal laminations in open domains in \({\mathbb{R}}^3\) with the singular set of C ¹-convergence being any properly embedded C ^1,1-curve. By Meeks’ C ^1,1-regularity theorem, the singular set of convergence of a Colding–Minicozzi limit minimal lamination \({\mathcal{L}}\) is a locally finite collection \(S({\mathcal{L}})\) of C ^1,1-curves that are orthogonal to the leaves of the lamination. Thus, our existence theorem gives a complete answer as to which curves appear as the singular set of a Colding–Minicozzi limit minimal lamination. In the case the curve is the unit circle \({\mathbb{S}}^1(1)\) in the (x ₁, x ₂)-plane, the classical Björling theorem produces an infinite sequence of complete minimal annuli H _n of finite total curvature which contain the circle. The complete minimal surfaces H _n contain embedded compact minimal annuli \(\overline{H}_n\) in closed compact neighborhoods N _n of the circle that converge as \(n \to \infty\) to \(\mathbb {R}^3 - x_3\) -axis. In this case, we prove that the \(\overline{H}_n\) converge on compact sets to the foliation of \(\mathbb {R}^3 - x_3\) -axis by vertical half planes with boundary the x ₃-axis and with \({\mathbb{S}}^1(1)\) as the singular set of C ¹-convergence. The \(\overline{H}_n\) have the appearance of highly spinning helicoids with the circle as their axis and are named bent helicoids.     

Porous medium equation and fast diffusion equation as gradient systems

We show that the Porous Medium Equation and the Fast Diffusion Equation, \(\dot u - \Delta {u^m} = f\), with m ∈ (0, ∞), can be modeled as a gradient system in the Hilbert space H ^?1(Ω), and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets Ω ? ? ⁿ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.     

Completely reducible super-simple designs with block size four and related super-simple packings

A design is said to be super-simple if the intersection of any two blocks has at most two elements. A super-simple design \({\mathcal{D}}\) with point set X, block set \({\mathcal{B}}\) and index λ is called completely reducible super-simple (CRSS), if its block set \({\mathcal{B}}\) can be written as \({\mathcal{B}=\bigcup_{i=1}^{\lambda} \mathcal{B}_i}\), such that \({\mathcal{B}_i}\) forms the block set of a design with index unity but having the same parameters as \({\mathcal{D}}\) for each 1 ≤ i ≤ λ. It is easy to see, the existence of CRSS designs with index λ implies that of CRSS designs with index i for 1 ≤ i ≤ λ. CRSS designs are closely related to q-ary constant weight codes (CWCs). A (v, 4, q)-CRSS design is just an optimal (v, 6, 4)_q+1 code. On the other hand, CRSS group divisible designs (CRSSGDDs) can be used to construct q-ary group divisible codes (GDCs), which have been proved useful in the constructions of q-ary CWCs. In this paper, we mainly investigate the existence of CRSS designs. Three neat results are obtained as follows. Firstly, we determine completely the spectrum for a (v, 4, 3)-CRSS design. As a consequence, a class of new optimal (v, 6, 4)₄ codes is obtained. Secondly, we give a general construction for (4, 4)-CRSSGDDs with skew Room frames, and prove that the necessary conditions for the existence of a (4, 2)-CRSSGDD of type g ^u are also sufficient except definitely for \({(g,u)\in \{(2,4),(3,4),(6,4)\}}\). Finally, we consider the related optimal super-simple (v, 4, 2)-packings and show that such designs exist for all v ≥ 4 except definitely for \({v\in \{4,5,6,9\}}\).     

Existence of solutions for a nonlinear elliptic problem of fourth order with weight

In this work we study the existence and regularity of solutions of the equation Δ _p ² u = λm|u| ^q?2 u with the boundary conditions of Navier in the case p ≠ q.