Nonexistence Results Research Articles

Existence and multiplicity of solutions for a class of (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Kirchhoff system with combined nonlinearities on graphs

By using the well-known mountain-pass theorem and Ekeland’s variational principle, we prove that there exist at least two fully nontrivial solutions for a (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Kirchhoff elliptic system with the Dirichlet boundary conditions and perturbation terms on a locally weighted and connected finite graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G=(V,E)$\\end{document}. We also present a necessary condition of the existence of semitrivial solutions for the system. Moreover, by using Ekeland’s variational principle and Clark’s Theorem, respectively, we prove that the system has at least one or multiple semitrivial solutions when the perturbation terms satisfy different assumptions. Finally, we present a nonexistence result of solutions.

Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products

Abstract Our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions. Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker spacetimes such as the Einstein-de Sitter spacetime, are also given. Furthermore, we study the nonexistence of entire solutions to the mean curvature flow equation. Our approach is based on a suitable conformal change of metric jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al.

On Small-Amplitude Asymmetric Water Waves

AbstractWe generalize the method used by Mæhlen and Seth (Asymmetric traveling wave solutions of the capillary–gravity Whitham equation, 2023, arxiv:2312.15343) used to prove the existence of small-amplitude asymmetric solutions to the capillary–gravity Whitham equation, so that it can be applied directly to a class of similar equations. The purpose is to prove or disprove the existence of asymmetric waves for the water wave problem or other model equations for water waves. Our main result in this paper is a theorem that gives both necessary and sufficient conditions for the existence of small-amplitude periodic asymmetric solutions for this class of equations. The result is then applied to an infinite depth capillary–gravity Whitham equation and an infinite depth capillary–gravity Babenko equation to show a nonexistence result of small-amplitude waves for these equations. This example also highlights the similarities between these equations suggesting the potential existence of small-amplitude asymmetric waves for the finite depth capillary–gravity Babenko equation.

Nontrivial solutions for multiparameter k-Hessian systems

This paper is devoted to studying the coupled system of multiparameter k-Hessian equations where B = {x ∈ ℝ n : |x| < 1}, λ1, λ2 are positive parameters, ρi (i = 1, 2) is singular near the boundary ∂B, f i(i = 1, 2) satisfies a combined superlinear condition at ∞. Employing the fixed point theorem of Krasnosel’skii type in Banach space, the existence and multiplicity of nontrivial radial solutions are established. In particular, the dependence of the solutions on the parameters λ1, λ2 is also discussed. Finally, we study the nonexistence results of nontrivial radial solutions for the case λ1, λ2 ≫ 1.

Liouville theorem of the regional fractional Lane–Emden equations

The purpose of this paper is to study the Liouville property for the Lane–Emden equation involving the regional fractional Laplacian ( − Δ ) Ω s u + Vu = h 1 u p + h 2 in Ω , u = 0 on ∂Ω , where s ∈ ( 0 , 1 ) , p>0, h 1 , h 2 are nonnegative functions and Ω ⊂ R N − 1 × [ 0 , + ∞ ) with N ≥ 2 , is an unbounded domain satisfying Ω t := { x ′ ∈ R N − 1 : ( x ′ , t ) ∈ Ω } with t ≥ 0 having an increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The potential V ( x ′ , t ) decays as t → + ∞ . The properties of the limit domain Ω ∞ := lim t → ∞ Ω t in R N − 1 play an important role to obtain the nonexistence of positive solutions for semilinear elliptic equations with the regional fractional Laplacian. For s ∈ ( 0 , 1 2 ] , we provide a surprising nonexistence result if Ω ∞ is bounded. This particular phenomenon occurs because of the peculiar properties of the regional fractional Laplacian with the order s ∈ ( 0 , 1 2 ] .

A Variational Model of Charged Drops in Dielectrically Matched Binary Fluids: The Effect of Charge Discreteness

This paper addresses the ill-posedness of the classical Rayleigh variational model of conducting charged liquid drops by incorporating the discreteness of the elementary charges. Introducing the model that describes two immiscible fluids with the same dielectric constant, with a drop of one fluid containing a fixed number of elementary charges together with their solvation spheres, we interpret the equilibrium shape of the drop as a global minimizer of the sum of its surface energy and the electrostatic repulsive energy between the charges under fixed drop volume. For all model parameters, we establish the existence of generalized minimizers that consist of at most a finite number of components “at infinity”. We also give several existence and non-existence results for classical minimizers consisting of only a single component. In particular, we identify an asymptotically sharp threshold for the number of charges to yield existence of minimizers in a regime corresponding to macroscopically large drops containing a large number of charges. The obtained non-trivial threshold is significantly below the corresponding threshold for the Rayleigh model, consistently with the ill-posedness of the latter and demonstrating a particular regularizing effect of the charge discreteness. However, when a minimizer does exist in this regime, it approaches a ball with the charge uniformly distributed on the surface as the number of charges goes to infinity, just as in the Rayleigh model. Finally, we provide an explicit solution for the problem with two charges and a macroscopically large drop.

A non-existence result for higher-order Hardy-Hénon inequality on punctured balls

A non-existence result for higher-order Hardy-Hénon inequality on punctured balls

Different aspects of blow-up property for a nonlinear wave equation

In this work, we are concerned with a nonlinear wave equation with variable exponents. In the presence of the logarithmic nonlinear source, we established a global nonexistence result with negative initial data and without imposing the Sobolev Logarithmic Inequality. The blow-up time is established with upper bound and lower bound. In addition, under some conditions on the initial data and for a specific class of relaxation functions, we established an infinite time blow-up result.

A nonlocal elliptic system with nonlinear singular terms

In this work we deal with the nonlocal elliptic system: 0 & \\hbox{in}\\ \\Omega, \\end{cases} \\] ]]> ( S ) { ( − Δ ) s 1 u = θ v q u 1 − θ in Ω , ( − Δ ) s 2 v = q v q − 1 u θ in Ω , u = v = 0 in R N ∖ Ω , u , v > 0 in Ω , where Ω ⊂ R N is a bounded regular domain ( C 2 is sufficient), 2s_2 $ ]]> N > 2 s 2 , s 1 , s 2 ∈ ( 0 , 1 ) with s 2 ≠ s 1 , 0 < θ < 1 and q>0. This work extends previous results obtained in the local case ( s 1 = s 2 = 1 ) see Boccardo L, Orsina L. [A variational semilinear singular system. Nonlinear Anal Theory Methods Appl. 2011;74(12):3849–3860. doi: 10.1016/j.na.2011.01.017]. Under some suitable conditions on the parameters on s 1 , s 2 , θ and q, we obtain the existence results by using approximation methods, variational techniques and the classical minimization, a nonexistence result has also been treated.

Competition with exclusive contracts in vertically related markets: An equilibrium non-existence result

I study a model in which two upstream firms compete to supply a homogeneous input to two downstream firms selling differentiated products. Upstream firms offer exclusive, discriminatory, public, two-part tariff contracts to the downstream firms. I show that, under very general conditions, this game does not have a pure-strategy subgame-perfect equilibrium. The intuition is that variable parts in such an equilibrium would have to be pairwise-stable; however, with pairwise-stable variable parts, downstream competitive externalities are not internalized, implying that upstream firms can profitably deviate. I contrast this non-existence result with earlier papers that found equilibria in related models.

Nonexistence Results for the Elliptic Equations with Fractional Laplacian and Variable Exponents Nonlinearities

Nonexistence Results for the Elliptic Equations with Fractional Laplacian and Variable Exponents Nonlinearities

Existence and non-existence results for a singular second-order elliptic equation

Using the variational methods, this paper is concerned with the existence and non-existence results to a class of elliptic equation with singular term on compact Riemannian manifold. In the context of complete, non-compact and non parabolic manifold, we obtain the non-existence result of solution for this type of elliptic problems.

Nonexistence results for a time-fractional biharmonic diffusion equation

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ∂tαu+∂tβu+uxxxx=h(t,x)|u|p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\partial _{t}^{\\alpha }u+\\partial _{t}^{\\beta }u+u_{xxxx}=h(t,x)|u|^{p}$\\end{document} in (0,∞)×(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(0,\\infty )\ imes (0,1)$\\end{document} subject to the initial conditions u(0,x)=u0(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(0,x)=u_{0}(x)$\\end{document}, ut(0,x)=u1(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u_{t}(0,x)=u_{1}(x)$\\end{document} and the Navier boundary conditions u(t,1)=uxx(t,1)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u(t,1)=u_{xx}(t,1)=0$\\end{document}, where α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha \\in (0,1)$\\end{document}, β∈(1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\beta \\in (1,2)$\\end{document}, ∂tα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\partial _{t}^{\\alpha}$\\end{document} (resp. ∂tβ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\partial _{t}^{\\beta}$\\end{document}) is the fractional derivative of order α (resp. β) with respect to the time-variable in the Caputo sense, p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p>1$\\end{document} and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator ∂4∂x4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\frac{\\partial ^{4}}{\\partial x^{4}}$\\end{document}, the domain, the initial conditions and the boundary condition, a general nonexistence result is established. Next, some special cases of weight functions h are discussed.

Post-Lie algebra structures for perfect Lie algebras

We study the existence of post-Lie algebra structures on pairs of Lie algebras ( g , n ) , where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect non-semisimple, and n is s l 3 ( C ) . We also show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect and n is reductive with a 1-dimensional center.

Positive solution for a singular and nonlocal problem of the N-Kirchhoff type

We investigate the existence of a positive solution for a singular elliptic problem of the N-Kirchhoff type involving a nonlocal operator. The nonlinearity considered in the equation consists of two terms: one has a subcritical, critical or supercritical exponential growth governed by the Trudinger–Moser inequality, and the other one presents a singularity at the origin. We also show a result of nonexistence of solutions in dimension 2. Finally, we study the asymptotic behavior of the solutions with respect to the parameter.

Semilinear elliptic problems via the nonlinear Rayleigh quotient with two nonlocal nonlinearities

It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: { − Δ u + V ( x ) u = μ ( I α 1 ∗ | u | q ) | u | q − 2 u − λ ( I α 2 ∗ | u | p ) | u | p − 2 u in R N , u ∈ H 1 ( R N ) , where p > q , λ , μ > 0 , α 1 ≤ α 2 ; α 1 , α 2 ∈ ( 0 , N ) , N ≥ 3 ; p ∈ ( 2 α 2 , 2 α 2 ∗ ) ; q ∈ ( 2 α 1 , 2 α 1 ∗ ) , 2 α j = ( N + α j ) / N and 2 α j ∗ = ( N + α j ) / ( N − 2 ) , j = 1 , 2 . Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp μ n > 0 and λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each μ > μ n where λ ∈ ( 0 , min ( λ ∗ , λ ∗ ) ) . The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each μ < μ n and λ > 0 .

Complex translation methods and its application to resonances for quantum walks

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier.

On noncompact warped product Ricci solitons

AbstractThe goal of this paper is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented as a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.

Calabi-Bernstein type results for critical points of a weighted area functional in [formula omitted] and [formula omitted]

In this paper we prove some Calabi-Bernstein type and non-existence results concerning complete [φ,e→3]-minimal surfaces in R3 whose Gauss maps lie on compacts subsets of open hemispheres of S2. We also give a general non-existence result for complete spacelike [φ,e→3]-maximal surfaces in L3 and, in particular, we obtain a Calabi-Bernstein type result when φ˙ is bounded.

Non-existence results for fourth order Hardy–Hénon equations in dimensions 2 and 3

We consider the fourth order Hardy–Hénon equationΔ2u=|x|σupin Rn with n≥2, σ>−4, and p>1. This is the fourth order analogy of the second order equation −Δu=|x|σup, known as the Hardy–Hénon equation, which was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above fourth order equation, the assumption n≥4 is often assumed. In this work, we are interested in classical solutions to the equation in the case of low dimensions, namely 2≤n≤3. Here by a classical solution u we mean u belongs to the class C(Rn)∩C4(Rn∖{0}) if σ<0 and C4(Rn) if σ≥0. We show that if 2≤n≤3 then the equation admits no classical solution. In fact, we are able to provide a single treatment for 2≤n≤4.