Nonexistence Results Research Articles

Constructions of difference sets in nonabelian 2-groups

Difference sets have been studied for more than 80 years. Techniques from algebraic number theory, group theory, finite geometry, and digital communications engineering have been used to establish constructive and nonexistence results. We provide a new theoretical approach which dramatically expands the class of $2$-groups known to contain a difference set, by refining the concept of covering extended building sets introduced by Davis and Jedwab in 1997. We then describe how product constructions and other methods can be used to construct difference sets in some of the remaining $2$-groups. We announce the completion of ten years of collaborative work to determine precisely which of the 56,092 nonisomorphic groups of order 256 contain a difference set. All groups of order 256 not excluded by the two classical nonexistence criteria are found to contain a difference set, in agreement with previous findings for groups of order 4, 16, and 64. We provide suggestions for how the existence question for difference sets in $2$-groups of all orders might be resolved.

Nonexistence results of solutions of parabolic-type equations and systems on the Heisenberg group

In this paper, we present necessary conditions for the existence of weak solution of the parabolic-type equations and systems on the Heisenberg group. The main technique for proving the results relies on the method of test functions.

Nonexistence of radial time-periodic solutions of reaction-diffusion equations with generic nonlinearities

We consider reaction-diffusion equations ut=Δu+f(u) on the entire space RN, N≥4. Assuming that the function f is sufficiently smooth (C2 is sufficient) and has only nondegenerate zeros, we prove that the equation has no bounded solutions u(x,t) which are radial in x, and periodic and nonconstant in t. We also prove some weaker nonexistence results for N=3. In dimensions N=1,2, the nonexistence of time-periodic solutions (radial or not) is known by results of Gallay and Slijepčević.

New results on permutation binomials of finite fields

After a brief review of the existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then focus on PBs of Fq2 of the form Xn(Xd(q−1)+a), where n and d are positive integers and a∈Fq2⁎. Our contributions include two nonexistence results: (1) If q is even and sufficiently large and aq+1≠1, then Xn(X3(q−1)+a) is not a PB of Fq2. (2) If 2≤d|q+1, q is sufficiently large and aq+1≠1, then Xn(Xd(q−1)+a) is not a PB of Fq2 under certain additional conditions. (1) partially confirms a recent conjecture by Tu et al. (2) is an extension of a previous result with n=1.

Existence and nonexistence of positive radial solutions of a quasilinear Dirichlet problem with diffusion

In this paper existence and nonexistence results of positive radial solutions of a Dirichlet m-Laplacian problem with different weights and a diffusion term inside the divergence of the form (a(|x|)+g(u))−γ, with γ>0 and a, g positive functions satisfying natural growth conditions, are proved. Precisely, we obtain a new critical exponent mα,β,γ⁎, which extends the one relative to case with no diffusion and it divides existence from nonexistence of positive radial solutions. The results are obtained via several tools such as a suitable modification of the celebrated blow up technique, Liouville type theorems, a fixed point theorem and a Pohožaev-Pucci-Serrin type identity.

On the Brezis–Nirenberg problem for the (p, q)-Laplacian

We prove some existence and nonexistence results for a class of critical (p, q)-Laplacian problems in a bounded domain. Our results extend and complement those in the literature for model cases.

Degree 2 Boolean Functions on Grassmann Graphs

We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are known for $n \geq 5$. This paper focusses on providing a list of examples on the case $d=2$ in general dimension and in particular for $(n, k)=(6,3)$ and $(n,k) = (8, 4)$.We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, $q$-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.

Topology of Asymptotically Conical Calabi–Yau and G2 Manifolds and Desingularization of Nearly Kähler and Nearly G2 Conifolds

A natural approach to the construction of nearly G_2 manifolds lies in resolving nearly G_2 spaces with isolated conical singularities by gluing in asymptotically conical G_2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G_2 manifolds, whose endpoint is the original nearly G_2 conifold and whose parameter is the scale of the glued in asymptotically conical G_2 manifold. We show that in many cases such a curve does not exist. The non-existence result is based on a topological result for asymptotically conical G_2 manifolds: if the rate of the metric is below -3, then the G_2 4-form is exact if and only if the manifold is Euclidean mathbb R^7. A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi–Yau 6-manifolds: if the rate of the metric is below -3, then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean mathbb R^6.

Horizon area bound and MOTS stability in locally rotationally symmetric solutions

In this paper, we study the stability of marginally outer trapped surfaces (MOTSs), foliating horizons of the form , embedded in locally rotationally symmetric class II perfect fluid spacetimes. An upper bound on the area of stable MOTS is obtained. It is shown that any stable MOTS of the types considered in these spacetimes must be strictly stably outermost, that is, there are no MOTS ‘outside’ of and homologous to . Aspects of the topology of the MOTS, as well as the case when an extension is made to imperfect fluids, are discussed. Some non-existence results are also obtained. Finally, the ‘growth’ of certain matter and curvature quantities on certain unstable MOTS are provided under specified conditions.

Compact Minimal Submanifolds in a Large Class of Riemannian Manifolds

Through a new technique, we provide uniqueness, rigidity and non-existence results for compact minimal submanifolds of arbitrary dimension in a large class of Riemannian manifolds, which include between others, Riemannian double-twisted and warped products. Moreover, we show that our results can be applied in particular, to space forms and Cartan–Hadamard manifolds, re-obtaining several classic results in a different approach. Interesting applications to Geometric Analysis are also showed.

Nonlinear perturbations of BLMP and YTSF equations

Some nonlinear evolutive equations of Mathematical Physics present infinitely many solutions described in many paper by different methods. For example Körteweg De Vries equation and Kadomtsev - Petviashvili equation are completely integrable despite the presence of quasilinear terms. In the present paper we perturb these kind of equations by positive nonlinear terms having polynomial growth. Assuming that the quasilinear term in the original equation has divergence form, we may apply test function method and establish a range of exponents for the perturbation so that a non-existence result of global weak solutions holds. Concerning initial data condition, an important difference with other equations studied by similar methods (wave, Tricomi and so on) appears. Indeed, we present a class of quasilinear equations for which the sign assumption on the initial data can be omitted and non-existence results still hold. Our basic examples are the perturbation of Boiti Leon Manna Pempinelli equation and Yu Toda Sasa Fukuyama equation. Finally we suggest open problems for other equations and other kind of perturbations.

Entire subsolutions of a kind of $ \boldsymbol{k} $-Hessian type equations with gradient terms

In this paper, we consider a $ k $-Hessian type equations $ S_k^{\frac{1}{k}}[D^2u+\mu|D u|I] = f(u) $ in $ \mathbb{R}^n $, and provide a necessary and sufficient condition for the solvability of entire admissible subsolutions, which can be regarded as a generalized Keller-Osserman condition. The existence and nonexistence results are proved in different ranges of the parameter $ \mu $ respectively, which embrace the standard Hessian equation case ($ \mu = 0 $) by Ji and Bao (Proc. Amer. Math. Soc. 138: 175–188, 2010) as a typical example. The difference between the semilinear case ($ k = 1 $) and the fully nonlinear case ($ k\ge 2 $) is also concerned.

Global existence and nonexistence of solutions for semilinear wave equation with a new condition

In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition on the source term $ f(u) $ such as \begin{document}$ \begin{equation*} \alpha \int_0^{u } f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma, \end{equation*} $\end{document} for some positive constants $ \alpha $, $ \beta $, and $ \sigma $, where $ \alpha >2 $ and $ \beta < \frac{\lambda_1(\alpha -2)}{2} $ with $ \lambda_1 $ being a first eigenvalue of Dirichlet-Laplacian. The condition on $ f(u) $ with $ \beta = 0 $ and $ \sigma = 0 $ was studied in [13] and [3]. By introducing a family of potential wells, we are able to establish the invariant sets, vacuum isolation of solutions, and a threshold result of global existence and nonexistence of solutions with the new condition on a source term. Moreover, we discuss the global existence and nonexistence of solutions for problem with initial condition $ E(0)<d $ and critical initial condition $ E(0) = d $.

On the critical behavior for time-fractional reaction diffusion problems

We first investigate the existence and nonexistence of weak solutions to the time-fractional reaction diffusion problem \begin{document}$ \frac{\partial^\alpha u}{\partial t^\alpha}-\frac{\partial^2 u}{\partial x^2}+u\geq x^{-a}|u|^p, \, \, t>0, \, x\in (0, 1], \quad u(0, x) = u_0(x), \, \, x\in (0, 1] $\end{document} under the inhomogeneous Dirichlet boundary condition \begin{document}$ u(t, 1) = \delta, \quad t>0, $\end{document} where $ u = u(t, x) $, $ 0<\alpha<1 $, $ \frac{\partial^\alpha }{\partial t^\alpha} $ is the time-Caputo fractional derivative of order $ \alpha $, $ a\geq 0 $, $ p>1 $ and $ \delta>0 $. We show that, if $ a\leq 2 $, the existence holds for all $ p>1 $ while if $ a>2 $, then the dividing line with respect to existence or nonexistence is given by the critical exponent $ p^* = a-1 $. The proof of the nonexistence result is based on nonlinear capacity estimates specifically adapted to the nonlocal nature of the problem, the modified Helmholtz operator $ -\frac{\partial^2}{\partial x^2}+I $, and the considered boundary condition. The existence part is proved by the construction of explicit solutions. We next extend our study to the case of systems.

Existence and nonexistence results for fifth-order multi-point boundary value problems involving integral boundary condition

Existence and nonexistence results for fifth-order multi-point boundary value problems involving integral boundary condition

VARIATIONAL FORMULATION FOR THE STURM-LIOUVILLE PROBLEM OF FRACTIONAL DIFFERENTIAL EQUATION WITH GENERALIZED (<i>P</i>, <i>Q</i>)-LAPLACIAN OPERATOR

In this paper, the Sturm-Liouville boundary value problem is studied for fractional differential equation with generalized (<i>p</i>, <i>q</i>)-Laplacian operator. By imposing mild assumptions on nonlinearity <i>f</i>, several new existence results of at least one or two nontrivial weak solutions are established through variational methods and critical point theorems. Furthermore, the criteria is also investigated for the nonexistence result.

Nonexistence results for nonlinear fractional differential inequalities involving weighted fractional derivatives

In this paper, we study the nonexistence of global solutions for certain classes of nonlinear fractional differential inequalities involving weighted fractional derivatives and a singular potential function. Namely, using the test function method with a judicious choice of the test function, we obtain sufficient criteria depending on the parameters of the considered problems, under which we have absence of global solutions. Next, some special cases of the potential function are discussed.

There is no Enriques surface over the integers

We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite étale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine and Abrashkin for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on counting rational points, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and Shepherd-Barron, some recent results on the base of their versal deformation, Shioda's theory of Mordell--Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.

A Nonexistence Result for Choquard-Type Hamiltonian System

In this article, we establish a nonexistence result of nontrivial non-negative solutions for the following Choquard-type Hamiltonian system by the Pohožaev identity , when , , , , , and , where and denotes the convolution in .

Qualitative analysis on the diffusive Holling–Tanner predator–prey model

We consider the diffusive Holling–Tanner predator–prey model subject to the homogeneous Neumann boundary condition. We first apply Lyapunov function method to prove some global stability results of the unique positive constant steady-state. And then, we derive a non-existence result of positive non-constant steady-states by a novel approach that can also be applied to the classical Sel'kov model to obtain the non-existence of positive non-constant steady-states if 0 &lt; p ≤ 1 .