Nonexistence Results Research Articles

On the existence of balanced metrics on six-manifolds of cohomogeneity one

We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced SU(n)-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull–Strominger system. In this paper, we provide a non-existence result for balanced non-Kähler text {SU}(3)-structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.

On a power-type coupled system with mean curvature operator in Minkowski space

We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space {M(u)+vα=0in B,M(v)+uβ=0in B,u|∂B=v|∂B=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} \\mathcal{M}(u)+ v^{\\alpha }=0\\quad \\text{in } B, \\\\ \\mathcal{M}(v)+ u^{\\beta }=0\\quad \\text{in } B, \\\\ u|_{\\partial B}=v|_{\\partial B}=0, \\end{cases} $$\\end{document} where mathcal{M}(w)=operatorname{div} ( frac{nabla w}{sqrt{1-|nabla w|^{2}}} ) and B is a unit ball in mathbb{R}^{N} (Ngeq 2). We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.

On the asymptotic Plateau problem in [formula omitted

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space SL˜2(R) with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in SL˜2(R) contained between two bounded entire minimal graphs separated by vertical distance less than 1+4τ2π has multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in SL˜2(R).

Characterizations of the $G_2(4)$ and $L_3(4)$ Near Octagons

A triple $(\mathcal{S},S,\mathcal{Q})$ consisting of a near polygon $\mathcal{S}$, a line spread $S$ of $\mathcal{S}$ and a set $\mathcal{Q}$ of quads of $\mathcal{S}$ is called a polygonal triple if certain nice properties are satisfied, among which there is the requirement that the point-line geometry $\mathcal{S}'$ formed by the lines of $S$ and the quads of $\mathcal{Q}$ is itself also a near polygon. This paper addresses the problem of classifying all near polygons $\mathcal{S}$ that admit a polygonal triple $(\mathcal{S},S,\mathcal{Q})$ for which a given generalized polygon $\mathcal{S}'$ is the associated near polygon. We obtain several nonexistence results and show that the $G_2(4)$ and $L_3(4)$ near octagons are the unique near octagons that admit polygonal triples whose quads are isomorphic to the generalized quadrangle $W(2)$ and whose associated near polygons are respectively isomorphic to the dual split Cayley hexagon $H^D(4)$ and the unique generalized hexagon of order $(4,1)$.

The critical exponent for semilinear [formula omitted]-evolution equations with a strong non-effective damping

In this paper, we find the critical exponent for the existence of global small data solutions to: utt+(−Δ)σu+(−Δ)θ2ut=f(u,ut),t≥0,x∈Rn,(u,ut)(0,x)=(0,u1(x)),in the case of so-called non-effective damping, θ∈(σ,2σ], where σ≠1 and f=|u|α or f=|ut|α, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers α>α̃ and do not exist, in general, for subcritical powers 1<α<α̃. Assuming initial data to be small in L1 or in some other Lp space, p∈(1,2), in addition to the energy space, the critical exponent only depends on the ratio n/(σp). We also prove the global existence of small data solutions in high space dimension for α>ᾱ, but we leave open to determine if a counterpart nonexistence result for α<ᾱ holds or not.

Existence and nonexistence of entire k-convex radial solutions to Hessian type system

In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire k-convex radial solutions is established by the monotone iterative method. Moreover, a nonexistence result is also obtained.

On tilings of asymmetric limited-magnitude balls

We study whether an asymmetric limited-magnitude ball may tile Zn. This ball generalizes previously studied shapes: crosses, semi-crosses, and quasi-crosses. Such tilings act as perfect error-correcting codes in a channel which changes a transmitted integer vector in a bounded number of entries by limited-magnitude errors.A construction of lattice tilings based on perfect codes in the Hamming metric is given. Several non-existence results are proved, both for general tilings, and lattice tilings. A complete classification of lattice tilings for two certain cases is proved.

Global bifurcation for N-dimensional p-Laplacian problem and its applications

In this paper, we are concerned with the global bifurcation results for quasilinear elliptic problem where λ is a parameter, . Let B be a unit open ball of with a smooth boundary . We shall show that there are two distinct unbounded continua and , consisting of the bifurcation branch if f is not necessarily differentiable at the origin with respect to , and there are two distinct unbounded continua and , consisting of the bifurcation branch if f is not necessarily differentiable at infinity with respect to . As the applications of the above result, we shall prove more details about the existence and multiplicity results of sign-changing solutions for the elliptic problem where and g is not necessarily differentiable at the origin and infinity with respect to . Furthermore, by using a comparison theorem, we also obtain a non-existence result of nodal solutions to the above problem.

A note on gradient Ricci soliton warped metrics

AbstractIn this note, we prove triviality and nonexistence results for gradient Ricci soliton warped metrics. The proofs stem from the construction of gradient Ricci solitons that are realized as warped products, from which we know that the base spaces of these products are Ricci–Hessian type manifolds. We study this latter class of manifolds as the most appropriate setting to prove our results.

On regular systems of finite classical polar spaces

Let P be a finite classical polar space of rank d. An m-regular system with respect to (k−1)-dimensional projective spaces of P, 1≤k≤d−1, is a set R of generators of P with the property that every (k−1)-dimensional projective space of P lies on exactly m generators of R. Regular systems of polar spaces are investigated. Some non-existence results about certain 1-regular systems of polar spaces with low rank are proved and a procedure to obtain m′-regular systems from a given m-regular system is described. Finally, three different construction methods of regular systems w.r.t. points of various polar spaces are discussed.

Global nonexistence and blow-up results for a quasi-linear evolution equation with variable-exponent nonlinearities

"In this paper, we consider a class of quasi-linear parabolic equations with variable exponents, $$a\left( x,t\right) u_{t}-\Delta _{m\left( .\right) }u=f_{p\left( .\right)}\left( u\right)$$ in which $f_{p\left( .\right)}\left( u\right)$ the source term, $a(x,t)&gt;0$ is a nonnegative function, and the exponents of nonlinearity $m(x)$, $p(x)$ are given measurable functions. Under suitable conditions on the given data, a finite-time blow-up result of the solution is shown if the initial datum possesses suitable positive energy, and in this case, we precise estimate for the lifespan $T^{\ast }$ of the solution. A blow-up of the solution with negative initial energy is also established."

Existence and nonexistence results for a class of Hamiltonian Choquard-type elliptic systems with lower critical growth on ℝ2

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem\[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \]when$(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$(if$N\geq 3$) and$(N+\alpha )/p + (N+\beta )/q \geq 2N$(if$N=2$), where$I_{\alpha }$and$I_{\beta }$denote the Riesz potential. Second, via variational methods and the generalized Nehari manifold, we show the existence of a nontrivial non-negative solution or a Nehari-type ground state solution for the problem\[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \]where$\alpha ,\,\beta \in (0,\,2)$and$f,\,g$have exponential critical growth in the Trudinger–Moser sense.

A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

We extend work by Christian et al. [Review of Economic Design 2007] on lobbying in multiple referenda by first providing a more fine-grained analysis of the computational complexity of the NP-complete Lobbying problem. Herein, given a binary matrix, the columns represent issues to vote on and the rows correspond to voters making a binary vote on each issue. An issue is approved if a majority of votes has a 1 in the corresponding column. The goal is to get all issues approved by modifying a minimum number of rows to all-1-rows. In our multivariate complexity analysis, we present a more holistic view on the nature of the computational complexity of Lobbying, providing both (parameterized) tractability and intractability results, depending on various problem parameterizations to be adopted. Moreover, we show non-existence results concerning efficient and effective preprocessing for Lobbying and introduce natural variants such as Restricted Lobbying and Partial Lobbying.

Nonexistence of solutions for indefinite fractional parabolic equations

We study fractional parabolic equations with indefinite nonlinearities∂u∂t(x,t)+(−Δ)su(x,t)=x1up(x,t),(x,t)∈Rn×R, where 0<s<1 and 1<p<∞. We first prove that all positive bounded solutions are monotone increasing along the x1 direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce several new ideas and developed a systematic approach which may also be applied to investigate qualitative properties of solutions for many other fractional parabolic problems.

Existence and nonexistence in the liquid drop model

We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei.

Nonexistence results for hyperbolic type inequalities involving the Grushin operator in exterior domains

We study the hyperbolic type differential inequality $$ u_{tt}(t,x,y)-\mathcal{L}_\ell u(t,x,y)\geq |u(t,x,y)|^p,\quad (t,x,y)\in (0,\infty)\times D_1\times D_2 $$ under the boundary conditions $$\displaylines{ u(t,x,y) \geq f(x),\quad (t,x,y)\in (0,\infty)\times \partial D_1\times D_2,\cr u(t,x,y) \geq g(y),\quad (t,x,y)\in (0,\infty)\times D_1\times \partial D_2, }$$ where \(p&gt;1\), \(D_k=\{z\in \mathbb{R}^{N_k}: |z|\geq 1\}\), \(k=1,2\), \(N_k\geq 2\), \(f\in L^1(\partial D_1)\), \(g\in L^1(\partial D_2)\), and \(\mathcal{L}_\ell\), \(\ell\in \mathbb{R}\), is the Grushin operator $$ \mathcal{L}_\ell u=\Delta_x u+ |x|^{2\ell} \Delta_y u. $$ We obtain sufficient conditions depending on \(p\), \(\ell\), \(N_1\), \(N_2\), \(f\), and \(g\), for which the considered problem admits no global weak solution. We discuss separately the four cases: \(N_1=N_2=2\); \(N_1=2\), \(N_2\geq 3\); \(N_1\geq 3\), \(N_2=2\); \(N_1,N_2\geq 3\).&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/75/abstr.html

Existence, Nonexistence and Multiplicity Results for Positive Radial Solutions of n−dimensional Elliptic System

Aims/ Objectives: In this paper, we study the existence, nonexistence and multiplicity of positive solutions to the n−dimensional elliptic system &#x0D; systems have been widely studied, but there is relatively little research on n-dimensional elliptic systems. We are very interested in this subject and want to study it. We give new conclusions on the existence, nonexistence and multiplicity of positive solutions for the n-dimensional elliptic system. &#x0D; Study Design: Study on the existence, nonexistence and multiplicity of positive solutions. Place and Duration of Study: School of Applied Science, Beijing Information Science &amp; Technology University, September 2019 to present. Methodology: We prove the existence, nonexistence and multiplicity of positive solutions by the results of fixed point index. Results: We give new conclusions of existence, nonexistence and multiplicity of positive solutionsfor the system. Conclusion: We prove the existence, nonexistence and multiplicity of positive solutions to the n-dimensional elliptic system &#x0D; and give new conclusions.

Normal conformal metrics on [formula omitted] with Q-curvature having power-like growth

Answering a question by M. Struwe [26] related to the blow-up behavior in the Nirenberg problem, we show that the prescribed Q-curvature equationΔ2u=(1−|x|p)e4u in R4,Λ:=∫R4(1−|x|p)e4udx<∞ has normal solutions (namely solutions which can be written in integral form, and hence satisfy Δu(x)=O(|x|−2) as |x|→∞) if and only if p∈(0,4) and(1+p4)8π2≤Λ<16π2.We also prove existence and non-existence results for the positive curvature case, namely for Δ2u=(1+|x|p)e4u in R4, and discuss some open questions.

On few-class Q-polynomial association schemes: feasible parameters and nonexistence results

We present the tables of feasible parameters of primitive 3 -class Q -polynomial association schemes and 4 - and 5 -class Q -bipartite association schemes (on up to 2800 , 10000 , and 50000 vertices, respectively), accompanied by a number of nonexistence results for such schemes obtained by analysing triple intersection numbers of putative open cases.

On the large time L∞-estimates of the Stokes semigroup in two-dimensional exterior domains

We prove that the Stokes semigroup is a bounded analytic semigroup on Lσ∞ of angle π/2 for two-dimensional exterior domains. This result is an end point case of the Lp-boundedness of the semigroup for p∈(1,∞), established by Borchers and Varnhorn (1993). The proof is based on the non-existence result of bounded steady flows (the Stokes paradox) and some asymptotic formula for the net force of the Stokes resolvent.