Lusternik Schnirelmann Category Research Articles

Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency

In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ > 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.

Multiplicity of Normalized Solutions to a Fractional Logarithmic Schrödinger Equation

We study the existence and multiplicity of normalized solutions to the fractional logarithmic Schrödinger equation (−Δ)su+V(ϵx)u=λu+ulogu2inRN, under the mass constraint ∫RN|u|2dx=a. Here, N≥2, a,ϵ>0, λ∈R is an unknown parameter, (−Δ)s is the fractional Laplacian and s∈(0,1). We introduce a function space where the energy functional associated with the problem is of class C1. Then, under some assumptions on the potential V and using the Lusternik–Schnirelmann category, we show that the number of normalized solutions depends on the topology of the set for which the potential V reaches its minimum.

Multiple Positive Solutions for Quasilinear Elliptic Problems in Expanding Domains

In this paper we prove the existence of multiple positive solutions for a quasilinear elliptic problem with unbalanced growth in expanding domains by using variational methods and the Lusternik–Schnirelmann category theory. Based on the properties of the category, we introduce suitable maps between the expanding domains and the critical levels of the energy functional related to the problem, which allow us to estimate the number of positive solutions by the shape of the domain.

A multiplicity result for a double perturbed Schrödinger-Bopp-Podolsky-Proca system

In this paper we will establish a multiplicity result for a double perturbed Schrödinger Bopp-Podolsky-Proca system on a compact 3-dimensional Riemannian manifold without boundary. Using the Lusternik-Schnirelmann Category, we will prove that the number of solutions of the system depends on the topological properties of the manifold.

Higher analogues of discrete topological complexity

In this paper, we introduce the nth discrete topological complexity and study its properties such as its relation with simplicial Lusternik–Schnirelmann category and how the higher dimensions of discrete topological complexity relate with each other. Moreover, we find a lower bound of n-th discrete topological complexity which is given by the nth usual topological complexity of the geometric realisation of that complex. Furthermore, we give an example for the strict case of that lower bound.

Multiple Normalized Solutions to a Choquard Equation Involving Fractional p-Laplacian in ℝN

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in RN. 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.

Multiple Normalized Solutions to a Logarithmic Schrödinger Equation via Lusternik–Schnirelmann Category

Multiple Normalized Solutions to a Logarithmic Schrödinger Equation via Lusternik–Schnirelmann Category

Multiple sign-changing solutions for superlinear (p,q)-equations in symmetrical expanding domains

In this paper we study quasilinear elliptic equations defined on symmetrical expanding domains driven by the (p,q)-Laplacian and with a superlinear right-hand side. Based on the Lusternik-Schnirelmann category we prove the existence of at least γ(Ωλ∖{0}) pairs (±u) of odd weak solutions with precisely two nodal domains, where γ stands for the genus.

Topological Complexity and LS-Category of Certain Manifolds

The Lusternik–Schnirelmann category and topological complexity are important invariants of topological spaces. In this paper, we calculate the Lusternik–Schnirelmann category and topological complexity of products of real projective spaces and their wedge products by using cup and zero-cup length. Also, we will find the topological complexity of R P 2 k + 1 by using the immersion dimension of R P 2 k + 1 .

Multiplicity of solutions to a nonlinear elliptic problem on a Riemannian orbifold

We employ the photography method to estimate the number of solutions to a nonlinear elliptic problem on a Riemannian orbifold in function of the Lusternik–Schnirelmann category of its submanifold of points with largest local group.

Different types of topological complexity based on higher homotopic distance

We first study the higher version of the relative topological complexity by using the homotopic distance. We also introduce the generalized version of the relative topological complexity of a topological pair with respect to both the Schwarz genus and the homotopic distance. With these concepts, we give some inequalities including the topological complexity and the Lusternik-Schnirelmann category, the most important parts of the study of robot motion planning in topology. Later, by defining the parametrized topological complexity via the homotopic distance, we present some estimates on the higher setting of this concept. Finally, we give some important examples of the parametrized topological complexities of fiber bundles with their fibers.

The p-widths of a surface

The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the p-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets.We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the p-widths of the round sphere are attained by lfloor sqrt{p}rfloor great circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be sqrt{pi }.En route to calculating the p-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero.

On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole \(\mathbb{R}^N\) via Lusternik-Schnirelmann Category

On Existence of Multiple Normalized Solutions to a Class of Elliptic Problems in Whole \(\mathbb{R}^N\) via Lusternik-Schnirelmann Category

G-category versus orbifold category

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.

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of the convolution type nonlinearity tends to the fractional critical one in the sense of Hardy–Littlewood–Sobolev inequality, we obtain the existence of multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, we prove that the topology of the domain furnishes a lower bound for the number of positive solutions.

Normal forms, invariant manifolds and Lyapunov theorems

<abstract><p>We present an approach to Lyapunov theorems about a center for germs of analytic vector fields based on the Poincaré–Dulac and Birkhoff normal forms. Besides new proofs of three Lyapunov theorems, we prove their generalization: if the Poincaré–Dulac normal form indicates the existence of a family of periodic solutions, then such a family really exists. We also present new proofs of Weinstein and Moser theorems about lower bounds for the number of families of periodic solutions; here, besides the normal forms, some topological tools are used, i.e., the Poincaré–Hopf formula and the Lusternik–Schnirelmann category on weighted projective spaces.</p></abstract>

Equivariant dimensions of groups with operators

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension $\mathsf{cd}\_G(\pi)$, the equivariant geometric dimension $\mathsf{cat}\_G(\pi)$, and the equivariant Lusternik–Schnirelmann category $\mathsf{gd}\_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\pi\rtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg–Ganea and Stallings–Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $\pi$ with $\mathsf{cat}\_G(\pi)=\mathsf{cd}\_G(\pi)=2$ and $\mathsf{gd}\_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings–Swan type result for families of subgroups which do not contain all finite subgroups.

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

Estimates of covering type and minimal triangulations based on category weight

Abstract In a recent publication [D. Govc, W. A. Marzantowicz and P. Pavešić, Estimates of covering type and the number of vertices of minimal triangulations, Discrete Comput. Geom. 63 2020, 1, 31–48], we have introduced a new method, based on the Lusternik–Schnirelmann category and the cohomology ring of a space X, that yields lower bounds for the size of a triangulation of X. In this current paper, we present an important extension that takes into account the fundamental group of X. In fact, if π 1 ⁢ ( X ) {\pi_{1}(X)} contains elements of finite order, then one can often find cohomology classes of high ‘category weight’, which in turn allow for much stronger estimates of the size of triangulations of X. We develop several weighted estimates and then apply our method to compute explicit lower bounds for the size of triangulations of orbit spaces of cyclic group actions on a variety of spaces including products of spheres, Stiefel manifolds, Lie groups and highly-connected manifolds.

On Lusternik–Schnirelmann category and topological complexity of non-k-equal manifolds

On Lusternik–Schnirelmann category and topological complexity of non-k-equal manifolds