Nonexistence Results Research Articles

Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

Small data scattering of 2d Hartree type Dirac equations

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity c(|⋅|−γ⁎〈ψ,βψ〉)βψ with c∈R∖{0}, 0<γ<2. Our aim is to show the small data global well-posedness and scattering in Hs for s>γ−1 and 1<γ<2. The difficulty stems from the singularity of the low-frequency part |ξ|−(2−γ)χ{|ξ|≤1} of potential. To overcome it we adapt Up−Vp space argument and bilinear estimates of [27,25] arising from the null structure. We also provide nonexistence result for scattering in the long-range case 0<γ≤1.

A nonexistence result for CMC surfaces in hyperbolic 3-manifolds

We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.

Existence and Nonexistence of Positive Solutions for Singular (p,\xa0q)-Equations with Superdiffusive Perturbation

We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is parametric and exhibits the combined effects of a singular term and of a superdiffusive one. We prove an existence and nonexistence result for positive solutions depending on the value of the parameter lambda in overset{circ }{{mathbb {R}}}_+=(0,+infty ).

Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

Abstract We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) , 2 s ⋆ := 2 ⁢ N N - 2 ⁢ s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝ N {\mathbb{R}^{N}} or a smooth symmetric bounded domain, and D 0 s ⁢ ( Ω ) {D^{s}_{0}(\Omega)} is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s. u s {u_{s}} converges to a l.e.s.s. u t {u_{t}} as s goes to any t &gt; 0 {t&gt;0} . In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t - ε {t-\varepsilon} . A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any s &gt; 1 {s&gt;1} .

Infinitely many sign-changing solutions of a critical fractional equation

In this paper, we obtain results of nonexistence of nonconstant positive solutions, and also existence of an unbounded sequence of sign-changing solutions for some critical problems involving conformally invariant operators on the unit sphere, in particular to the fractional Laplacian operator in the Euclidean space. Our arguments are based on a reduction of the initial problem in the Euclidean space to an equivalent problem on the standard unit sphere and vice versa, what together with blow up arguments, a variant of Pohozaev’s type identity, a refinement of regularity results for this type operators, and finally, by exploiting the symmetries of the sphere.

The axisymmetric σk-Nirenberg problem

We study the problem of prescribing σk-curvature for a conformal metric on the standard sphere Sn with 2≤k<n/2 and n≥5 in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function K near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of K of flatness order β∈[2,n). We prove among other things the following statements. (1) When β>n−2k, the solution set is compact, has a nonzero total degree counting and is therefore non-empty. (2) When β=n−2k, there is an explicit positive constant C(K) associated with K. If C(K)>1, the solution set is compact with a nonzero total degree counting and is therefore non-empty. If C(K)<1, the solution set is compact but the total degree counting is 0, and the solution set is sometimes empty and sometimes non-empty. (3) When 2n−2k≤β<n−2k, the solution set is compact, but the total degree counting is zero, and the solution set is sometimes empty and sometimes non-empty. (4) When β<n−2k2, there exists K for which there exists a blow-up sequence of solutions with unbounded energy. In this same range of β, there exists also some K for which the solution set is empty.

New nonexistence results on circulant weighing matrices

A weighing matrix W = (wi,j) is a square matrix of order n and entries wi,j in {0,± 1} such that WWT = kIn. In his thesis, Strassler gave a table of existence results for circulant weighing matrices with n ≤ 200 and k ≤ 100. In the latest version of Strassler’s table given by Tan, there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary searches outside Strassler’s table, and characterize the known proper circulant weighing matrices.

Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source

In this work, we study a plate equation with time delay in the velocity, frictional damping, and logarithmic source term. Firstly, we obtain the local and global existence of solutions by the logarithmic Sobolev inequality and the Faedo-Galerkin method. Moreover, we prove the stability and nonexistence results by the perturbed energy and potential well methods.

The Nirenberg problem of prescribed Gauss curvature on $S^2$

We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on S^{2} conformal to the round metric. A key tool is to employ the smooth Cheeger–Gromov compactness theorem to obtain general and essentially sharp a priori estimates for Gauss curvatures K contained in naturally defined stable regions. We prove that in such stable regions, the map u \to K_{g} , g = e^{2u}g_{+1} is a proper Fredholm map with well-defined degree on each component. This leads to a number of new existence and non-existence results. We also present a new proof and generalization of the Moser theorem on Gauss curvatures of even conformal metrics on S^{2} . In contrast to previous work, the work here does not use any of the Sobolev-type inequalities of Trudinger–Moser–Aubin–Onofri.

Global small data solutions for semilinear waves with two dissipative terms

In this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity |u|^p or nonlinearity of derivative type |u_t|^p, in any space dimension ngeqslant 1, for supercritical powers p>{bar{p}}. The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive L^r-L^q long time decay estimates for the solution in the full range 1leqslant rleqslant qleqslant infty . The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers p<{bar{p}}.

Positive solutions to superlinear semipositone problems on the exterior of a ball

We consider the problem where , n>2, λ is a positive parameter, belongs to a class of continuous functions which satisfy certain decay assumptions, and belongs to a class of continuous functions which are superlinear at ∞ with . Recently, several authors have studied positive radial solutions to this problem assuming that the weight function is radial. We allow non-radial weights and study the existence and non-existence of positive solutions. We prove the existence of a positive solution for small values of λ using variational methods. A non-existence result is established for large values of λ.

Existence and non-existence results for fractional Kirchhoff Laplacian problems

Existence and non-existence results for fractional Kirchhoff Laplacian problems

Liouville-Type Theorems for Sign-Changing Solutions to Nonlocal Elliptic Inequalities and Systems with Variable-Exponent Nonlinearities

We consider the fractional elliptic inequality with variable-exponent nonlinearity $$\begin{aligned} (-\Delta )^{\frac{\alpha }{2}} u+\lambda \, \Delta u \ge |u|^{p(x)}, \quad x\in {\mathbb {R}}^N, \end{aligned}$$where \(N\ge 1\), \(\alpha \in (0,2)\), \(\lambda \in {\mathbb {R}}\) is a constant, \(p: {\mathbb {R}}^N\rightarrow (1,\infty )\) is a measurable function, and \((-\Delta )^{\frac{\alpha }{2}}\) is the fractional Laplacian operator of order \(\frac{\alpha }{2}\). A Liouville-type theorem is established for the considered problem. Namely, we obtain sufficient conditions under which the only weak solution is the trivial one. Next, we extend our study to systems of fractional elliptic inequalities with variable-exponent nonlinearities. Besides the consideration of variable-exponent nonlinearities, the novelty of this work consists in investigating sign-changing solutions to the considered problems. Namely, to the best of our knowledge, only nonexistence results of positive solutions to fractional elliptic problems were investigated previously. Our approach is based on the nonlinear capacity method combined with a pointwise estimate of the fractional Laplacian of some test functions, which was derived by Fujiwara (Math Methods Appl Sci 41:4955–4966, 2018) [see also Dao and Reissig (A blow-up result for semi-linear structurally damped \(\sigma \)-evolution equations, arXiv:1909.01181v1, 2019)]. Note that the standard nonlinear capacity method cannot be applied to the considered problems due to the change of sign of solutions.

Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems

This article deals with the study of the following singular quasilinear equation: $$\begin{aligned} (P) \left\{ \ -\Delta _{p}u -\Delta _{q}u = f(x) u^{-\delta },\; u>0 \text { in }\; \Omega ; \; u=0 \text { on } \partial \Omega , \right. \end{aligned}$$where \(\Omega \) is a bounded domain in \({\mathbb {R}}^N\) with \(C^2\) boundary \(\partial \Omega \), \(1< q< p<\infty \), \(\delta >0\) and \(f\in L^\infty _{loc}(\Omega )\) is a non-negative function which behaves like \(\text {dist}(x,\partial \Omega )^{-\beta },\) \(\beta \ge 0\) near the boundary of \(\Omega \). We prove the existence of a weak solution in \(W^{1,p}_{loc}(\Omega )\) and its behaviour near the boundary for \(\beta <p\). Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case \(\beta <2-\frac{1}{p}\). Subsequently, for the case \(\beta \ge p\), we prove the non-existence result. Moreover, we prove Hölder regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see (1.6)). This result is completely new and of independent interest. In addition to this, we prove Hölder regularity of minimal weak solutions of (P) for the case \(\beta +\delta \ge 1\) that has not been fully answered in former contributions even for p-Laplace operators.

Nonexistence results for parabolic equations involving the $${\varvec{p}}$$-Laplacian and Hardy–Leray-type inequalities on Riemannian manifolds

Nonexistence results for parabolic equations involving the $${\varvec{p}}$$-Laplacian and Hardy–Leray-type inequalities on Riemannian manifolds

Fair and Efficient Allocations under Lexicographic Preferences

Envy-freeness up to any good (EFX) provides a strong and intuitive guarantee of fairness in the allocation of indivisible goods. But whether such allocations always exist or whether they can be efficiently computed remains an important open question. We study the existence and computation of EFX in conjunction with various other economic properties under lexicographic preferences--a well-studied preference restriction model in artificial intelligence and economics. In sharp contrast to the known results for additive valuations, we not only prove the existence of EFX and Pareto optimal allocations, but in fact provide an algorithmic characterization of these two properties. We also characterize the mechanisms that are, in addition, strategyproof, non-bossy, and neutral. When the efficiency notion is strengthened to rank-maximality, we obtain non-existence and computational hardness results, and show that tractability can be restored when EFX is relaxed to another well-studied fairness notion called maximin share guarantee (MMS).

A SAT-based Resolution of Lam's Problem

In 1989, computer searches by Lam, Thiel, and Swiercz experimentally resolved Lam's problem from projective geometry—the long-standing problem of determining if a projective plane of order ten exists. Both the original search and an independent verification in 2011 discovered no such projective plane. However, these searches were each performed using highly specialized custom-written code and did not produce nonexistence certificates. In this paper, we resolve Lam's problem by translating the problem into Boolean logic and use satisfiability (SAT) solvers to produce nonexistence certificates that can be verified by a third party. Our work uncovered consistency issues in both previous searches—highlighting the difficulty of relying on special-purpose search code for nonexistence results.

Spin(7) Instantons and Hermitian Yang\u2013Mills Connections for the Stenzel Metric

We use the highly symmetric Stenzel Calabi–Yau structure on T^{star }S^{4} as a testing ground for the relationship between the Spin(7) instanton and Hermitian–Yang–Mills (HYM) equations. We reduce both problems to tractable ODEs and look for invariant solutions. In the abelian case, we establish local equivalence and prove a global nonexistence result. We analyze the nonabelian equations with structure group SO(3) and construct the moduli space of invariant Spin(7) instantons in this setting. This is comprised of two 1-parameter families—one of them explicit—of irreducible Spin(7) instantons. Each carries a unique HYM connection. We thus negatively resolve the question regarding the equivalence of the two gauge theoretic PDEs. The HYM connections play a role in the compactification of this moduli space, exhibiting a removable singularity phenomenon that we aim to further examine in future work.

Nonexistence of Subcritical Solitary Waves

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to “half-solitary” waves (e.g. bores) which decay in only one direction.