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In this paper, we study the existence of multiple normalized solutions to the following Schrödinger–Poisson system with general nonlinearities: \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+\phi u=f(u)+\lambda u & \hbox{in $\mathbb{R}^3$,} \\ -\varepsilon^2\Delta\phi=u^2& \hbox{in $\mathbb{R}^3$,}\\ \int_{\mathbb{R}^3}|u|^2{\rm d}x=\varepsilon^3 a^2,\ \end{cases} \end{equation*} where $\varepsilon$, $a>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V(x):\mathbb{R}^3 \rightarrow [0,\infty)$ is a continuous function, and $f$ is a differentiable function satisfying $L^2$-subcritical growth. Through using the minimization techniques and the Lusternik–Schnirelmann category, we prove that the numbers of normalized solutions are related to the topology of the set where the potential $V(x)$ attains its minimum value.

This article is concerned with the existence of multiple normalized solutions for a class of Choquard equations with a parametric perturbation −ε2Δu+V(x)u=λu+ε−α(Iα*F(u))f(u),x∈RN,∫RN∣u∣2dx=a2εN,\left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V\left(x)u=\lambda u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }* F\left(u))f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2}{\varepsilon }^{N},\hspace{1.0em}& \end{array}\right. where a>0a\gt 0 is a constant, ε>0\varepsilon \gt 0 is a parameter, N≥3N\ge 3, α∈(0,N)\alpha \in \left(0,N), λ∈R\lambda \in {\mathbb{R}} is unknown and appears as a Lagrange multiplier, ff is a continuous function with L2{L}^{2}-subcritical growth, and V:RN→[0,∞)V:{{\mathbb{R}}}^{N}\to \left[0,\infty ) is a continuous function, satisfying del Pino and Felmer’s local conditions. With the help of the penalization method, and Lusternik-Schnirelmann theory, we investigate the relationship between the number of positive normalized solutions and the topology of the set, where the potential VV attains its minimum value if the parameter ε>0\varepsilon \gt 0 is small.

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in R N . With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem. [ABSTRACT FROM AUTHOR]

In this study, we compute some examples of sectional category secat(f) and sectional number sec(f) for continuous maps f related to finite spaces. Moreover, we introduce an invariant secatk(f) for a map f between finite spaces using the k-th barycentric subdivision and show the equality secatk(f) = secat(B(f)) for sufficiently large k, where B(f) is the induced map on the associated polyhedra. [ABSTRACT FROM AUTHOR]

Over a local ring R, the theory of cohomological support varieties attaches to any bounded complex M of finitely generated R-modules an algebraic variety {\mathrm {V}}_R(M) that encodes homological properties of M. We give lower bounds for the dimension of {\mathrm {V}}_R(M) in terms of classical invariants of R. In particular, when R is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M has finite projective dimension, we also give an upper bound for \dim {\mathrm {V}}_R(M) in terms of the dimension of the radical of the homotopy Lie algebra of R. This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R. Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring. [ABSTRACT FROM AUTHOR]

We consider the Lorentz force equation d d t (m x ˙ 1 − | x ˙ | 2 / c 2 ) = q (E (t , x) + x ˙ × B (t , x)) , x ∈ R 3 , in the physically relevant case of a singular electric field E. Assuming that E and B are T -periodic in time and satisfy suitable further conditions, we prove the existence of infinitely many T -periodic solutions. The proof is based on a min-max principle of Lusternik-Schnirelmann type, in the framework of non-smooth critical point theory. Applications are given to the problem of the motion of a charged particle under the action of a Liénard-Wiechert potential and to the relativistic forced Kepler problem. [ABSTRACT FROM AUTHOR]

For simplicial complexes and simplicial maps, the notion of being in the same contiguity class is defined as the discrete version of homotopy. In this paper, we study the contiguity distance, SD, between two simplicial maps adapted from the homotopic distance. In particular, we show that simplicial versions of LS -category and topological complexity are particular cases of this more general notion. Moreover, we present the behaviour of SD under the barycentric subdivision, and its relation with strong collapsibility of a simplicial complex. [ABSTRACT FROM AUTHOR]

We consider a geodesic problem in a manifold endowed with a Randers–Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo’s navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik–Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution. [ABSTRACT FROM AUTHOR]

In this paper we show how the notion of the Lusternik–Schnirelmann relative category can be used to study a multiplicity problem for brake orbits in a potential well which is homeomorphic to the N-dimensional unit disk. The estimate of the relative category of the set of chords with endpoints on the (N − 1)-unit sphere was shown to the third author by Fadell and Husseini while he was visiting the University of Wisconsin at Madison. [ABSTRACT FROM AUTHOR]

We are trying to look over the Lusternik–Schnirelmann theory (L-S theory, for short) and the Topological Complexity (TC, for short) as a natural extension of the L-S theory. In particular, we focus on the impact of the ideas originated from E. Fadell and S. Husseini on both theories. More precisely, we see how their ideas on a category weight and a relative category drive the L-S theory and the TC. [ABSTRACT FROM AUTHOR]

We present a comparative study of certain invariants defined for group actions and their analogues defined for orbifolds. In particular, we prove that Fadell’s equivariant category for G-spaces coincides with the Lusternik–Schnirelmann category for orbifolds when the group is finite. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the Lusternik– Schnirelmann category and the topological complexity of two closed oriented manifolds connected by a degree one map. [ABSTRACT FROM AUTHOR]

Let Gk;n be the Grassmann manifold of k-planes in Rn+k. The Lusternik-Schnirelmann category and topological complexity are important invariants of topological spaces. In this note we calculate the Lusternik-Schnirelmann category and topological complexity of certain products of Grassmannian manifolds by using cup and zero-cup length. Also we will find the lower and upper bounds of the topological complexity of some Grassmannian manifolds by the same method. [ABSTRACT FROM AUTHOR]

In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler–Laplacian operator, we introduce faithful dimension pairs by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik–Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry–Émery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik–Schnirelmann dimension pair. [ABSTRACT FROM AUTHOR]

Abstract We prove the existence of multiple positive solutions for a fractional Laplace problem with critical growth and sign-changing weight in non-contractible domains.

The concept of the homotopic distance and its higher analogs are introduced in [7]. In this paper we introduce some important properties of higher homotopic distances, investigate the conditions under which cat, secat and higher dimensional topological complexity categories are equal to the higher homotopic distance, and give alternative proofs, using higher homotopic distances, to some TCn-related theorems. [ABSTRACT FROM AUTHOR]

We use the homotopy invariance of equivariant principal bundles to prove that the equivariant A-category of Clapp and Puppe is invariant under Morita equivalence. As a corollary, we obtain that both the equivariant Lusternik-Schnirelmann category of a group action and the invariant topological complexity are invariant under Morita equivalence. This allows a definition of topological complexity for orbifolds. [ABSTRACT FROM AUTHOR]

Let Sp (n) be the symplectic group of quaternionic (n × n) -matrices. For any 1 ≤ k ≤ n , an element A of Sp (n) can be decomposed in A = [ α T β P ] with P a (k × k) -matrix. In this work, starting from a singular value decomposition of P , we obtain what we call a relative singular value decomposition of A. This feature is well adapted for the study of the quaternionic Stiefel manifold X n , k , and we apply it to the determination of the Lusternik-Schnirelmann category of Sp (k) in X 2 k − j , k , for j = 0 , 1 , 2. [ABSTRACT FROM AUTHOR]

We show a method to determine topological complexity from the fibrewise view point, which provides an alternative proof for tc (K) = 4 , where K denotes Klein bottle. [ABSTRACT FROM AUTHOR]

Abstract In this article we study the higher topological complexity TC r (X) in the case when X is an aspherical space, X = K (π , 1) and r ≥ 2. We give a characterisation of TC r (K (π , 1)) in terms of classifying spaces for equivariant Bredon cohomology. Our recent paper [7] , joint with M. Grant and G. Lupton, treats the special case r = 2. We also obtain in this paper useful lower bounds for TC r (π) in terms of cohomological dimension of subgroups of π × π × ... × π (r times) with certain properties. As an illustration of the main technique we find the higher topological complexity of Higman's group. We also apply our method to obtain a lower bound for the higher topological complexity of the right angled Artin (RAA) groups, which, as was established in [16] by a different method (in a more general situation), coincides with the precise value. We finish the paper by a discussion of the TC -generating function ∑ r = 1 ∞ TC r + 1 (X) x r encoding the values of the higher topological complexity TC r (X) for all values of r. We show that in many examples (including the case when X = K (H , 1) with H being a RAA group) the TC -generating function is a rational function of the form P (x) (1 − x) 2 where P (x) is an integer polynomial with P (1) = cat (X). [ABSTRACT FROM AUTHOR]

Abstract In [11] , a new approximating invariant TC D for topological complexity was introduced called D -topological complexity. In this paper, we explore more fully the properties of TC D and the connections between TC D and invariants of Lusternik–Schnirelmann type. We also introduce a new TC -type invariant TC ˜ that can be used to give an upper bound for TC , (1) TC (X) ≤ TC D (X) + TC ˜ (X). This then entails a connectivity-dimension estimate TC (X) ≤ TC D (X) + ⌈ 2 dim X − k k + 1 ⌉ , where X is a finite dimensional simplicial complex with k -connected universal cover X ˜. The above inequality is a refinement of an estimate given by Dranishnikov [5]. [ABSTRACT FROM AUTHOR]

Abstract Let (M , g) be any closed Riemannianan manifold and (N , h) be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product (M × N , g + δ h) has at least Cat (M) + 1 solutions for δ small enough, where Cat (M) denotes the Lusternik–Schnirelmann-category of M. The solutions obtained are functions of M and C a t (M) of them have energy arbitrarily close to the minimum. [ABSTRACT FROM AUTHOR]

The categorical sequence of a space X is a sequence of integers that encodes the growth of the Lusternik-Schnirelmann category of its CW skeleta as dimension increases. Restrictions on these sequences found in [13] have proven to be powerful tools in studying and computing L-S category, motivating the search for additional restrictions. In this paper we study the initial three-term segments of the categorical sequences of rational spaces of finite type. We show that there is another restriction: a sequence of the form (a; b; a + b; …) is the categorical sequence of a rational space of finite type if and only if b &38801; 2 mod a - 1. With the possible exception of a small number of values of c for each a, all other three-term initial sequences are realizable by simply-connected rational spaces of finite type. [ABSTRACT FROM AUTHOR]

Abstract The notion of a dual polyhedral product is introduced as a generalization of Hovey's definition of Lusternik–Schnirelmann cocategory. Properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if X is a simply-connected space then the weak cocategory of X equals the homotopy nilpotency class of Ω X. This answers a fifty year old problem posed by Ganea. The methods are applied to determine the homotopy nilpotency class of quasi- p -regular exceptional Lie groups and sporadic p -compact groups. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and the multiplicity of positive solutions for the following class of fractional Schrödinger equations ϵ 2 s ( − Δ ) s u + V ( x ) u = f ( u ) in R N , where ϵ > 0 is a parameter, s ∈ ( 0 , 1 ) , N > 2 s , V : R N → R is a continuous positive potential, and f : R → R is a C 1 superlinear nonlinearity which does not satisfy the Ambrosetti–Rabinowitz condition. The main result is established by using minimax methods and Ljusternik–Schnirelmann theory of critical points. [ABSTRACT FROM AUTHOR]

Abstract For a pair of spaces X and Y such that Y ⊆ X , we define the relative topological complexity of the pair (X , Y) as a new variant of relative topological complexity. Intuitively, this corresponds to counting the smallest number of motion planning rules needed for a continuous motion planner from X to Y. We give basic estimates on the invariant, and we connect it to both Lusternik–Schnirelmann category and topological complexity. In the process, we compute this invariant for several example spaces including wedges of spheres, topological groups, and spatial polygon spaces. In addition, we connect the invariant to the existence of certain types of axial maps. [ABSTRACT FROM AUTHOR]

In this paper, we define the digital Lusternik-Schnirelmann category catκ, introduce some of its properties, and discuss how the adjacency relation affects the digital Lusternik-Schnirelmann category. [ABSTRACT FROM AUTHOR]

We study a nonlinear Choquard equation with weighted terms and critical Sobolev-Hardy exponent. We apply variational methods and Lusternik-Schnirelmann category to prove the multiple positive solutions for this problem. [ABSTRACT FROM AUTHOR]

This paper presents a notion of Lusternik–Schnirelmann category for small categories, which is an invariant under homotopy equivalences based on natural transformations. We focus on the relationship between this categorical Lusternik–Schnirelmann category and the classical one via the classifying space. We provide a combinatorial method to calculate the classical Lusternik–Schnirelmann category of the classifying space of a finite acyclic category, taking the barycentric subdivision into account. Moreover, we establish the product inequality for fibered and cofibered categories as an analogue of the inequality of the classical Lusternik–Schnirelmann category for Hurewicz fibrations. [ABSTRACT FROM AUTHOR]

The Cayley transform for orthogonal groups is a well known construction with applications in real and complex analysis, linear algebra and computer science. In this work, we construct Cayley transforms on Stiefel manifolds. Applications to the Lusternik–Schnirelmann category and optimization problems are presented. [ABSTRACT FROM AUTHOR]

The configuration space of n labeled disks of radius r inside the unit disk is denoted Conf n , r ( D 2 ) . We study how the cohomology of this space depends on r . In particular, given a cohomology class of Conf n , 0 ( D 2 ) , for which r does its restriction to Conf n , r ( D 2 ) vanish? A related question: given the configuration space Seg n , r ( D 2 ) of n labeled, oriented segments of length r , it has a map to ( S 1 ) n that records the direction of each segment. For which r does this angle map have a continuous section? The paper consists of a collection of partial results, and it contains many questions and conjectures. [ABSTRACT FROM AUTHOR]

Given a map f : M → N of degree 1 of closed manifolds, is it true that the Lusternik–Schnirelmann category of the range of the map is not more than the category of the domain? We discuss this and some related questions. [ABSTRACT FROM AUTHOR]

In this paper, we study the Lusternik–Schnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Several properties of this new invariant are shown, including its relevance to simplicial products and fibrations. We relate this category of a map to the classical Lusternik–Schnirelmann category of a map between finite topological spaces. Finally, we show how the simplicial category of a map may be used to define and study a simplicial version of the category weight of Y. Rudyak and J. Strom. [ABSTRACT FROM AUTHOR]

We show that the Lusternik-Schnirelmann category of the homotopy cofiber of the diagonal map of non-orientable surfaces equals three. Also, we prove that the topological complexity of non-orientable surfaces of genus > 4 is four. [ABSTRACT FROM AUTHOR]

Solutions of the stationary semilinear Cahn-Hilliard-type equation which are exponentially decaying at infinity, are studied. Using the mounting pass lemma allows us to determinate the existence of a radially symmetric solution. On the other hand, the application of Lusternik-Schnirel'man (L-S) category theory shows the existence of, at least, a countable family of solutions. However, through numerical methods it is shown that the whole set of solutions, even in 1D, is much wider. This suggests that, actually, there exists, at least, a countable set of countable families of solutions, in which only the first one can be obtained by the L-S min-max approach. [ABSTRACT FROM AUTHOR]

We first compute James’ sectional category (secat) of the Ganea map gk of any map ιX in terms of the sectional category of ιX: we show that secat gk is the integer part of secat ιX/(k+1). Next we compute the relative category (relcat) of gk. In order to do this, we introduce the relative category of order k (relcatk) of a map and show that relcat gk is the integer part of relcatkιX/(k+1). Then we establish some inequalities linking secat and relcat of any order: we show that secat ιX⩽relcatkιX⩽secat ιX+k+1 and relcatkιX⩽relcatk+1ιX⩽relcatkιX+1. We give examples that show that these inequalities may be strict. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new monadic structure on the intermediate constructions, TnF, of Goodwillie's calculus of functors. We show that as a result these functors take values in spaces of Hopkins' symmetric Lusternik-Schnirelmann (LS) cocategory ≤ n, which is an upper bound on the homotopy nilpotence class of the space. This property allows us to extend results of Biedermann-Dwyer linking Goodwillie calculus to homotopy nilpotence and of Chorny-Scherer on the vanishing of Whitehead products for spaces which are values of n-excisive functors. We also use a dual form of our adjunction to give a rigorous formulation of homotopy functor analog of McCarthy's Dual Calculus, where n-co-excisive functors take certain pullback cubes to pushout cubes, and dualize our results of calculus and LS cocategory to dual calculus and LS category. [ABSTRACT FROM AUTHOR]

We study the behavior of the abstract sectional category in the Quillen, the Strøm and the Mixed proper model structures on topological spaces and prove that, under certain reasonable conditions, all of them coincide with the classical notion. As a result, the same conclusions hold for the abstract Lusternik–Schnirelmann category and the abstract topological complexity of a space. [ABSTRACT FROM AUTHOR]

Félix, Halperin, and Lemaire have shown that the rational module category and the rational Toomer invariant coincide for simply connected Poincaré duality complexes. We establish an analogue of this result for the sectional category of a fibration. [ABSTRACT FROM AUTHOR]

In this paper we establish a natural definition of Lusternik–Schnirelmann category for simplicial complexes via the well known notion of contiguity. This simplicial category has the property of being invariant under strong equivalences, and it only depends on the simplicial structure rather than its geometric realization. In a similar way to the classical case, we also develop a notion of simplicial geometric category. We prove that the maximum value over the simplicial homotopy class of a given complex is attained in the core of the complex. Finally, by means of well known relations between simplicial complexes and posets, specific new results for the topological notion of LS-category are obtained in the setting of finite topological spaces. [ABSTRACT FROM AUTHOR]

We show that the topological complexity of an aspherical space X is bounded below by the cohomological dimension of the direct product A × B , whenever A and B are subgroups of π 1 ( X ) whose conjugates intersect trivially. For instance, this assumption is satisfied whenever A and B are complementary subgroups of π 1 ( X ) . This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zero-divisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group. [ABSTRACT FROM AUTHOR]