Ljusternik Schnirelmann Category Research Articles

Multiple positive solutions for a fractional elliptic system with critical nonlinearities

In this paper, we study the multiplicity results of positive solutions for a fractional elliptic system involving both concave-convex and critical growth terms. With the help of the Nehari manifold and the Ljusternik-Schnirelmann category, we prove that the problem admits at least $\operatorname{cat}(\Omega)+1$ distinct positive solutions.

Multiple positive solutions for Kirchhoff-type problems in $${\mathbb{R}^3}$$ R 3 involving critical Sobolev exponents

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent $$\left\{\begin{array}{ll}-\left(a+b\displaystyle\int\limits_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\right)\Delta u+u=\lambda f(x)u^{q-1}+g(x)u^5,\quad x\in \mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3), \end{array} \right.$$ where a, b > 0, 4 < q < 6, and \({\lambda}\) is a positive parameter. Under certain assumptions on f(x) and g(x) and \({\lambda}\) is small enough, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g. The Nehari manifold and Ljusternik–Schnirelmann category are the main tools in our study. Moreover, using the Mountain Pass Theorem, we give an existence result about \({\lambda}\) large.

Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions

In this article, we prove the existence and multiplicity of positive solutions for the following fractional elliptic equation with sign-changing weight functions: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u= a_\lambda(x)|u|^{q-2}u+b(x)|u|^{2^*_\alpha-1}u &{\rm in}\,\,\Omega, u=0\,\,&{\rm in}\,\,\R^N\setminus\Omega, \end{array} \right. \end{eqnarray*} where $0<\alpha<1$, $ \Omega $ is a bounded domain with smooth boundary in $ \R^N $ with $N>2\alpha$ and $ 2^*_\alpha=2N/(N-2\alpha)$ is the fractional critical Sobolev exponent. Our multiplicity results are based on studying the decomposition of the Nehari manifold and the Ljusternik-Schnirelmann category.

Multiple positive solutions for semi-linear elliptic systems involving sign-changing weight on manifolds with conical singularities

In this paper, we use the Nehari manifold method and Ljusternik-Schnirelmann category to prove the multiplicity result of positive solutions for the semi-linear elliptic systems with critical cone Sobolev exponent on manifolds with conical singularities.

A simple proof of multiplicity for periodic solutions of Lagrangian difference systems with relativistic operator and periodic potential

being a positive integer, T-periodic solutions of systems of difference equations in of the formare considered when is periodic in and in the . The approach is based upon a modification method and classical Ljusternik–Schnirelmann category.

Multiple positive solutions for a class of Kirchhoff type problems involving critical Sobolev exponents

In this paper, we study the multiplicity results of positive solutions for a class of Kirchhoff type problems involving critical growth terms. With the help of Nehari manifold and Ljusternik–Schnirelmann category theory, we investigate how the coefficient g(x) of the critical nonlinearity affects the number of positive solutions of that problem. Furthermore, we obtain a relationship between the number of positive solutions and the topology of the global maximum set of g.

Existence and multiplicity results for perturbed Kirchhoff-type Schrödinger systems in [formula omitted

In this paper we study a class of Kirchhoff-type Schrödinger systems with a positive parameter ε. By applying variational methods and the Ljusternik–Schnirelmann category theory, under some suitable conditions we obtain the existence and multiplicity results for the systems when the parameter ε is sufficiently small.

Multiple positive solutions for indefinite semilinear elliptic problems involving a critical Sobolev exponent

In this paper, we study the decomposition of the Nehari manifold by exploiting the combination of concave and convex nonlinearities. The result is subsequently used, in conjunction with the Ljusternik–Schnirelmann category and variational methods, to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical Sobolev exponent.

Multiplicity of positive solutions for a class of Schr&amp;ouml;dinger-Poisson systems with critical growth

We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Schrodinger-Poisson system where e > 0 is a parameter, V is a positive continuous potential bounded away from zero satisfying some conditions and f is a subcritical nonlinearity, 2 * = 6 is the critical Sobolev exponent in R 3 . We use a variational method to obtain the multiplicity of positive solutions by means of the Ljusternik-Schnirelmann category of the set where V attains its minimum value.

Periodic solutions for a class of nonlinear discrete Hamiltonian systems via critical point theory

In this paper, we study the multiple existence of periodic solutions of the discrete n-dimensional Hamiltonian system with V periodic both in t and the components of q. By showing that the action functional of this Hamiltonian system is invariant under the action of the non-compact group , we construct a new functional for which the Palais-Smale condition holds and prove multiple existence of periodic solutions of this system via Ljusternik–Schnirelmann category theory and Morse theory.

Multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent

In this paper, we study the decomposition of the Nehari manifold via the combination of concave and convex nonlinearities. Furthermore, we use this result and the Ljusternik–Schnirelmann category to prove that the existence of multiple positive solutions for a Dirichlet problem involving critical Sobolev exponent.

Semiclassical Limit For the Nonlinear Klein Gordon Equation in Bounded Domains

Abstract We are interested in the existence of standing waves for the nonlinear Klein Gordon equation ε2□ψ + Wʹ(ψ) = 0 in a bounded domain D. A standing wave has the form ψ(t, x) = u(x)e-iwt/ε; for these solutions the Klein Gordon equation becomes We want to use a Benci-Cerami type argument in order to prove a the existence of several standing waves localized in suitable points of D. The main result of this paper is that, under suitable growth condition on W, for ε suffciently small, we have at least cat(D) stationary solutions of equation (†), where cat(D) is the Ljusternik-Schnirelmann category. The proof is achieved by solving a constrained critical point problem via variational techniques.

Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik–Schnirelmann category and Morse theory.

Existence and multiplicity results for massive particles trajectories in a universe with boundary

In an open time-convex region Λ of a strongly causal Lorentzian manifold (M,g), we consider an event p and a timelike, injective curve γ. We look for geodesics connecting p and γ in Λ and satisfying the conservation law g(z)[ż,ż]=−E for a fixed E&amp;gt;0. It is already known that such geodesics are the stationary points of the arrival time functional τ. Our main result is to prove the existence of a decreasing flow for τ, by means of a shortening procedure. This makes possible to apply to τ global variational methods obtaining existence and multiplicity results (using the Ljusternik–Schnirelmann category theory) and also to develop a Morse theory.

Periodic Orbits and Subharmonics of Dynamical Systems on Non-Compact Riemannian Manifolds

where Dt(x* (t)) is the covariant derivative of x* along the direction of x* and {R the Riemannian gradient, has been studied when M is a noncontractible manifold (see [2, 8, 9, 14]), assuming, if M is non-compact, the existence of a function on M convex at infinity . When V is bounded the difficulties arise from the lack of compactness of M; indeed, in this case the action functional does not satisfy the Palais Smale compactness condition. On the other side, if V is unbounded the action functional is unbounded both from below and from above. Therefore neither min max methods and linking arguments can be used since the loop space is not linear nor the Ljusternik Schnirelmann category theory can be applied although M has a non-trivial topology. article no. 0136

Graded lie algebras of depth one

To each simply connected topological space is associated a graded Lie algebra; the rational homotopy Lie algebra. The Avramov-Felix conjecture says that for a space of finite Ljusternik-Schnirelmann category this Lie algebra contains a free Lie subalgebra on two generators. We prove the conjecture in the case when the Lie algebra has depth one.

A Note on the Category of the Free Loop Space

A useful result in critical point theory is that the LjusternikSchnirelmann category of the space of based loops on a compact simply connected manifold $M$ is infinite (because the cup length of $M$ is infinite). However, the space of free loops on $M$ may have trivial products. This note shows that, nevertheless, the space of the free loops also has infinite category.

Extensions of Ljusternik-Schnirelmann category theory to relative and equivariant theories with an application to an equivariant critical point theorem

We extend Ljusternik-Schnirelmann category theory to a relative G-category theory. The theory is related to the equivariant cohomological index of Fadell and Husseini. After giving computations for a number of examples, we present an application of the theory to the problem of finding critical points of invariant functionals.

A note on the category of the free loop space

A useful result in critical point theory is that the LjusternikSchnirelmann category of the space of based loops on a compact simply connected manifold M M is infinite (because the cup length of M M is infinite). However, the space of free loops on M M may have trivial products. This note shows that, nevertheless, the space of the free loops also has infinite category.

Minimal coverings of manifolds with balls

For a closed manifold M, denote by C(M) the minimal number of balls which suffice to cover M. It is shown that C(M) coincides with the Ljusternik-Schnirelmann category cat M if the latter is not too low compared with the dimension of M. In this case it follows in particular that C(M) is an invariant of the homotopy type of M. One of the applications of this result is the following: Let M be a closed manifold of sufficiently high category. Then cat(M×S1)=cat M+1. This is a partial affirmative answer to a long-standing conjecture.