Nonexistence Theorems Research Articles

Bi-$f$-Harmonic Legendre Curves on $(\alpha,\beta)$-Trans-Sasakian Generalized Sasakian Space Forms

In this study, we consider bi-$f$-harmonic Legendre curves on $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form. We provide the necessary and sufficient conditions for a Legendre curve to be bi-$f$-harmonic on $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form without any restrictions by a main theorem. Afterward, we investigate these conditions under nine different cases. As a result of these investigations, we obtain the original theorems and corollaries as well as the nonexistence theorems. We perform these investigations according to the $\rho_{2}$ and $\rho_{3}$ functions from the curvature tensor of the $(\alpha,\beta)$-trans-Sasakian generalized Sasakian space form, the curvature and torsion of the bi-$f$-harmonic Legendre curve, and finally, the positions of the basis vectors relative to each other.

Global existence and multiplicity of positive solutions for anisotropic eigenvalue problems

Abstract We consider an eigenvalue problem driven by the anisotropic (p, q)-Laplacian and with a Carathéodory reaction which is (p(z) − 1)-sublinear as x → + ∞. We look for positive solutions. We prove an existence, nonexistence and multiplicity theorem which is global in the parameter λ > 0, that is, we prove a bifurcation-type theorem which describes in an exact way the changes in the set of positive solutions as the parameter λ varies on ℝ̊+ = (0, + ∞).

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.

Nonexistence for fractional differential inequalities and systems in the sense of Erdélyi-Kober

<p>Nonexistence theorems constitute an important part of the theory of differential and partial differential equations. Motivated by the numerous applications of fractional differential equations in diverse fields, in this paper, we studied sufficient conditions for the nonexistence of solutions (or, equivalently, necessary conditions for the existence of solutions) for nonlinear fractional differential inequalities and systems in the sense of Erdélyi-Kober. Our approach is based on nonlinear capacity estimates specifically adapted to the Erdélyi-Kober fractional operators and some integral inequalities.</p>

The Morse Index of Sacks–Uhlenbeck α‐Harmonic Maps for Riemannian Manifolds

In this paper, first we prove a nonexistence theorem for α‐harmonic mappings between Riemannian manifolds. Second, the instability of nonconstant α‐harmonic maps is studied with regard to the Ricci curvature criterion of their codomain. Then, we estimate the Morse index for measuring the degree of instability of some particular α‐harmonic maps. Furthermore, the notion of α‐stable manifolds and its applications are considered. Finally, we investigate the α‐stability of any compact Riemannian manifolds admitting a nonisometric conformal vector field and any Einstein Riemannian manifold under certain assumptions on the smallest positive eigenvalue of its Laplacian operator on functions.

Sharp critical exponents for nonlinear equations with the fractional Laplacian

ABSTRACT In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely \\alpha \\end{align*} $$]]> ( − Δ ) α 2 ( u m ) = u | u | q − 1 + w ( x ) , x ∈ R N , 1 ≤ m < q , 0 < α ≤ 2 , N > α and \\alpha. \\end{align*} $$]]> ∂ k u ∂ t k + ( − Δ ) α 2 u = u q , ( x , t ) ∈ R N × ( 0 , + ∞ ) , k ≥ 1 , 0 < α ≤ 2 , N > α . Solutions defined for all x ∈ R N of the first equation are referred to as entire solutions, while solutions defined for all ( x , t ) ∈ R N × [ 0 , + ∞ ) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w.

Non-existence theorems on infinite order corks

Suppose that X,X′ are simply-connected closed exotic 4-manifolds. It is well-known that X′ is obtained by an order 2 cork twist of X. We give an infinite family of exotic 4-manifolds not generated by any infinite order cork. This is the first example admitting such a condition. We prove a necessary condition of 4-dimensional Ozsváth-Szabó invariants for a family to be generated by an infinite order cork and as an application give non-contractible relatively exotic 4-manifolds that are never induced by any cork. Furthermore, we give an estimate of the number of Ozsváth-Szabó invariants of 4-manifolds generated by a cork.

Semi-parallel Hopf real hypersurfaces in the complex quadric

In this paper, we introduce the new notion of semi-parallel real hypersurface in the complex quadric \(Q^{m}\). Moreover, we give a nonexistence theorem for semi-parallel Hopf real hypersurfaces in the complex quadric \(Q^{m}\) for \(m \geq 3\).

Existence and Nonexistence of Positive Solutions for Perturbations of the Anisotropic Eigenvalue Problem

We consider a Dirichlet problem, which is a perturbation of the eigenvalue problem for the anisotropic p-Laplacian. We assume that the perturbation is (p(z)−1)-sublinear, and we prove an existence and nonexistence theorem for positive solutions as the parameter λ moves on the positive semiaxis. We also show the existence of a smallest positive solution and determine the monotonicity and continuity properties of the minimal solution map.

On periodic Ambrosetti-Prodi-type problems

&lt;abstract&gt;&lt;p&gt;This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $ \lambda $. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.&lt;/p&gt;&lt;/abstract&gt;

On metrics projectively and holomorphically projectively equivalent to metrics of parabolic Riemannian and Kähler manifolds

We prove a number of nonexistence theorems for metrics projectively and holomorphically projectively equivalent to parabolic metrics and metrics of finite volume of complete Riemannian and K?hler manifolds, respectively.

The variation of a functional on the Riemannian submanifolds and its applications

The aim of this paper is to derive the first variation of a functional on an oriented submanifold in the Riemannian manifold involving an arbitrary vector field. After obtaining the first variation formula, a notion of $\sigma$-mean curvature is naturally introduced. We also find an important identity involving the $\sigma$-mean curvature and the divergence of the tangent component of the vector field. As an application, we get a general formula of the $\sigma$-mean curvature for the hypersurfaces in a class of Finsler manifolds called the general $(\alpha,\beta)$-manifolds (introduced in [38]). Hence the vanishing $\sigma$-mean curvature characterizes the minimal hypersurfaces under the Busemann-Hausdorff measure ([29]) and the Holmes-Thompson measure ([23]). We also give a general formula for the hypersurface in a Randers manifold involving the navigation data without any restriction on the vector field. In terms of the identity, we prove some nonexistence theorems of the closed orientable minimal submanifold in some special non-Minkowskian Finsler manifolds.

Nonexistence results for ultra-parabolic equations and systems involving fractional Laplacian operator in Heisenberg group

In this paper, sufficient conditions are obtained for the nonexistence of solutions to the Cauchy problems for ultra-parabolic equations involving fractional Laplacian operator. The nonexistence result is also extended to the corresponding system. The results to such problems are derived in the Heisenberg group. The proofs of our nonexistence theorems are based on the nonlinear capacity method. This method is based on the choice of a suitable test function in the weak formulation of the considered problems.

Inequalities for the Class of Warped Product Submanifold of Para-Cosymplectic Manifolds

The aim of this paper is to study the warped product pointwise semislant submanifolds in the para-cosymplectic manifold with the semi-Riemannian metric. For which, firstly we provide the more generalized definition of pointwise slant submanifolds and related characterization results followed by the definition of pointwise slant distributions and pointwise semislant submanifolds. We also derive some results for different foliations on distribution, and lastly, we defined pointwise semislant warped product submanifold, given existence and nonexistence results, basic lemmas, theorems, and optimal inequalities for the ambient manifold.

Some types of $f$-biharmonic and bi-$f$-harmonic curves

In this paper, we determine necessary and sufficient conditions for a non-Frenet Legendre curve to be $f$-harmonic, $f$-biharmonic, bi-$f$-harmonic, biminimal and $f$-biminimal in three-dimensional normal almost paracontact metric manifold. Besides, we obtain some nonexistence theorems.

The discrete Safronov-Dubovskiı̌ aggregation equation: Instantaneous gelation and nonexistence theorem

We consider the discrete Safronov-Dubovskiı̌ aggregation model in which monomers play a key role to form large clusters. It is proved that mass-loss phenomena can occur at any time interval for a specific aggregation rate. The case of nonexistence of solution is also studied.

Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds

We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our main result recovers some nonexistence theorems from the literature, established in the whole Euclidean space.

The nonexistence of gradient almost Ricci solitons warped product

In this paper, by slightly modifying Li-Yau's technique so that we can handle drifting Laplacians, we were able to find three different gradient estimates for the warping function, one for each sign of the Einstein constant of the fiber manifold. As an application, we exhibit a nonexistence theorem for gradient almost Ricci solitons possessing certain metric properties on the base of the warped product.

Nonexistence of supersymmetry breaking counterexamples to the Nelson-Seiberg theorem

Counterexample models to the Nelson-Seiberg theorem have been discovered, and their features have been studied in previous literature. All currently known counterexamples have generic superpotentials respecting the R-symmetry, and more R-charge 2 fields than R-charge 0 fields. But they give supersymmetric vacua with spontaneous R-symmetry breaking, thus violate both the Nelson-Seiberg theorem and its revisions. This work proves that the other type of counterexamples do not exist. When there is no R-symmetry, or there are no more R-charge 2 fields than R-charge 0 fields in models with R-symmetries, generic superpotentials always give supersymmetric vacua. There exists no specific arrangement of R-charges or non-R symmetry representations which makes a counterexample with a supersymmetry breaking vacuum. This nonexistence theorem contributes to a refined classification of R-symmetric Wess-Zumino models.

A Constructive Approach About the Existence of Positive Solutions for Minkowski Curvature Problems

In this paper, we give an existence theorem about positive solutions for the Dirichlet boundary value problem of one dimensional Minkowski curvature equations. We apply the theorem to one parameter family of problems to investigate a constructive method for numerical range of parameters where positive solutions exist. Moreover, we establish a nonexistence theorem of positive solutions for the corresponding one parameter family of problems. The coefficient function may be singular at the boundary and nonlinear term satisfies a sublinear growth condition. Main argument for the proof of existence theorem is employed by Krasnoselskii’s theorem of cone expansion and compression. We give a numerical algorithm and various examples to illustrate numerical information about ranges of the existence and nonexistence parameters which have been given only in a theoretical manner so far.