Lusternik-Schnirelman Theory Research Articles

Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik–Schnirelman theory, and new estimates on the category of suitable configuration spaces.

Semiclassical solutions for a critical Choquard–Poisson system with competitive potentials

In this article, we establish the existence, concentration behaviour and decay estimation of semiclassical ground state solutions for a critical Choquard–Poisson system with multiple competitive potentials by variational methods. Moreover, we study the multiplicity of solutions by using the Lusternik–Schnirelmann theory.

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

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.

Double bubbles with high constant mean curvatures in Riemannian manifolds

We obtain existence of double bubbles of large and constant mean curvatures in Riemannian manifolds. These arise as perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor. We also obtain general multiplicity results via Lusternik–Schnirelman theory, and extra ones in case of double bubbles whose opposite boundaries have the same mean curvature.

Existence of the Eigenvalues for the Cone Degenerate p-Laplacian

The present paper is concerned with the eigenvalue problem for cone degenerate p-Laplacian. First the authors introduce the corresponding weighted Sobolev s-paces with important inequalities and embedding properties. Then by adapting Lusternik-Schnirelman theory, they prove the existence of infinity many eigenvalues and eigenfunctions. Finally, the asymptotic behavior of the eigenvalues is given.

Ground state and multiple solutions for critical fractional Schrodinger-Poisson equations with perturbation terms

In this article, we study a class of critical fractional Schrodinger-Poisson system with two perturbation terms. By using variational methods and Lusternik-Schnirelman category theory, the existence of ground state and two nontrivial solutions are established.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/07/abstr.html

Multiple solutions for singularly perturbed nonlinear magnetic Schrödinger equations

In this paper we consider singularly perturbed nonlinear Schrödinger equations with electromagnetic potentials and involving continuous nonlinearities with subcritical, critical or supercritical growth. By means of suitable variational techniques, truncation arguments and Lusternik–Schnirelman theory, we relate the number of nontrivial complex-valued solutions with the topology of the set where the electric potential attains its minimum value.

Semiclassical states for Choquard type equations with critical growth: critical frequency case * *Minbo Yang is the corresponding author who is partially supported by NSFC (11971436, 12011530199) and ZJNSF (LD19A010001); Yanheng Ding is partially supported by NSFC (11871242); Fashun Gao is partially supported by NSFC (11901155).

In this paper we are interested in the existence of semiclassical states for the Choquard type equation where 0 < μ < N, N ⩾ 3, ɛ is a positive parameter and G is the primitive of g which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. The potential function V(x) is assumed to be nonnegative with V(x) = 0 in some region of , which means it is of the critical frequency case. Firstly, we study a Choquard equation with double critical exponents and prove the existence and multiplicity of semiclassical states by the mountain-pass lemma and the genus theory. Secondly, we consider a class of critical Choquard equation without lower perturbation, by establishing a global compactness lemma for the nonlocal Choquard equation, we prove the multiplicity of high energy semiclassical states by the Lusternik–Schnirelman theory.

Lusternik–Schnirelman theory for the action integral of the Lorentz force equation

In this paper we introduce new Lusternik–Schnirelman type methods for nonsmooth functionals including the action integral associated to the relativistic Lagrangian of a test particle under the action of an electromagnetic field $$\begin{aligned} {\mathcal {L}}(t,q,q')=1-\sqrt{1-|q'|^2}+q'\cdot W(t,q) - V(t,q), \end{aligned}$$where $$V:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ and $$W:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}^3$$ are two $$C^1$$-functions with V even and W odd in the second variable. By applying them, we obtain various multiplicity results concerning T-periodic solutions of the relativistic Lorentz force equation in Special Relativity, $$\begin{aligned} \left( \frac{q'}{\sqrt{1-|q'|^2}}\right) '=E(t,q) + q'\times B(t,q), \end{aligned}$$where $$ E=-\nabla _q V-\frac{\partial W}{\partial t}, B=\hbox {curl}_q\, W. $$ The zero Dirichlet boundary value conditions are considered as well.

Multiplicity of semiclassical states for Schrödinger–Poisson systems with critical frequency

We are concerned with the Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{lll} -\,\epsilon ^2\Delta u+V(x)u+K(x)\phi u=u^5,&{}\quad x\in {\mathbb {R}}^3,\\ -\,\Delta \phi =K(x)u^2, &{}\quad x\in {\mathbb {R}}^3, \end{array}\right. \end{aligned}$$where $$\epsilon >0$$ is a parameter, $$V\in L^{\frac{3}{2}}({\mathbb {R}}^3)$$ and $$K\in L^{2}({\mathbb {R}}^3)$$ are nonnegative functions and V is assumed to be zero in some region of $${\mathbb {R}}^3$$, which means it is of the critical frequency case. By virtue of a global compactness lemma and Lusternik–Schnirelman theory, we show the multiplicity of high energy semiclassical states.

Multiplicity of semiclassical states for fractional Schrödinger equations with critical frequency

We are concerned with the fractional Schrödinger equation ϵ2α(−Δ)αu+V(x)u=|u|2α∗−2u,x∈RN,where ϵ>0 is a parameter, 0<α<1, N≥3, 2α∗=2NN−2α, V∈LN2α(RN) is a nonnegative function and V is assumed to be zero in some region of RN, which means it is of the critical frequency case. By virtue of a global compactness lemma, two barycenter functions and Lusternik–Schnirelman theory, we show the multiplicity of high energy semiclassical states.

The effect of the domain topology on the number of solutions of fractional Laplace problems

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^n$$ is an open bounded set with continuous boundary, $$n>2s$$ with $$s\in (0,1),(-\Delta )^{s}$$ is the fractional Laplacian operator, $$\mu $$ is a positive real parameter, $$q\in [2, 2^*_s)$$ and $$2^*_s=2n/(n-2s)$$ is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of $$\Omega $$ . Precisely, we show that the problem has at least $$cat_{\Omega }(\Omega )$$ nontrivial solutions, provided that $$q=2$$ and $$n\geqslant 4s$$ or $$q\in (2, 2^*_s)$$ and $$n>2s(q+2)/q$$ , extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

A Generalized Palais-Smale Condition in the Fr\'{e}chet space setting

The Palais-Smale condition was introduced by Palais and Smale in the mid-sixties and applied to an extension of Morse theory to infinite dimensional Hilbert spaces. Later this condition was extended by Palais for the more general case of real functions over Banach-Finsler manifolds in order to obtain Lusternik-Schnirelman theory in this setting. Despite the importance of Fr\'{e}chet spaces, critical point theories have not been developed yet in these spaces.Our aim in this paper is to extend the Palais-Smale condition to the cases of $C^1$-functionals on Fr\'{e}chet spaces and Fr\'{e}chet-Finsler manifolds of class $C^1$. The difficulty in the Fr\'{e}chet setting is the lack of a general solvability theory for differential equations. This restricts us to adapt the deformation results (which are essential tools to locate critical points) as they appear as solutions of Cauchy problems. However, Ekeland proved the result, today is known as Ekleand’s variational principle, concerning the existence of almost-minimums for a wide class of real functions on complete metric spaces. This principle can be used to obtain minimizing Palais-Smale sequences. We use this principle along with the introduced conditions to obtain some customary results concerning the existence of minima in the Fr\'{e}chet setting.Recently it has been developed the projective limit techniques to overcome problems (such as solvability theory for differential equations) with Fr\'{e}chet spaces. The idea of this approach is to represent a Fr\'{e}chet space as the projective limit of Banach spaces. This approach provides solutions for a wide class of differential equations and every Fr\'{e}chet space and therefore can be used to obtain deformation results. This method would be the proper framework for further development of critical point theory in the Fr\'{e}chet setting.

Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$(-i\nabla - A(x))^2{u} = \mu|u|^{q-2}u + |u|^{2^*-2}u\, {\rm in}\, \Omega,\quad u=0\, {\rm on}\, \partial\, \Omega$$ where \({\Omega \subset \mathbb{R}^N(N \geq 4)}\) is a bounded domain with smooth boundary, \({A: \overline{\Omega} \rightarrow \mathbb{R}^N}\) is a continuous magnetic potential and \({2 \leq q < 2^* = \frac{2N}{N-2}}\). Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of Ω.

Multiple Solutions for a Quasilinear Schrödinger Equation on $\mathbb{R}^{N}$

The multiplicity of positive weak solutions is established for quasilinear Schrödinger equations −L p u+(λA(x)+1)|u| p−2 u=h(u) in $\mathbb{R}^{N}$ , where L p u=ϵ p Δ p u+ϵ p Δ p (u 2)u, A is a nonnegative continuous function and nonlinear term h has a subcritical growth. We achieved our results by using minimax methods and Lusternik-Schnirelman theory of critical points.

Multiplicity of solutions for a NLS equations with magnetic fields in $$\mathbb {R}^{N}$$ R N

We investigate the multiplicity of nontrivial weak solutions for a class of complex equations. This class of problems are related with the existence of solitary waves for a nonlinear Schodinger equation. The main result is established by using minimax methods and Lusternik–Schnirelman theory of critical points.

Multiplicity of nontrivial solutions to a biharmonic equation via Lusternik-Schnirelman theory

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of biharmonic problem Δ2u=μ|u|q−2u+|u|2蜧蜧−2uinΩ,u=∂u∂η=0on∂Ω, where Ω⊂RN is a bounded domain with smooth boundary. Using the Lusternik–Schnirelman theory, we relate the number of solutions with the topology of Ω. Copyright © 2012 John Wiley & Sons, Ltd.

A minimax theorem in the presence of unbounded Palais-Smale sequences

In a reflexive Banach space we consider a family of functionals that may admit unbounded Palais-Smale sequences. Under some structural conditions we provide a suitable Deformation Lemma that is obtained by modifying the classical pseudo-gradient flow. This leads to a variant of the minimax principle in the Lusternik-Schnirelman Theory.

On the Lusternik-Schnirelman theory of a real cohomology class

Farber developed a Lusternik-Schnirelman theory for finite CW-complexes X and cohomology classes ξ H1(X;ℝ). This theory has similar properties as the classical Lusternik-Schnirelman theory. In particular in [7] Farber defines a homotopy invariant cat(X,ξ) as a generalization of the Lusternik-Schnirelman category. If X is a closed smooth manifold this invariant relates to the number of zeros of a closed 1-form ω representing ξ. Namely, a closed 1-form ω representing ξ which admits a gradient-like vector field with no homoclinic cycles has at least cat(X,ξ) zeros. In this paper we define an invariant F(X,ξ) for closed smooth manifolds X which gives the least number of zeros a closed 1-form representing ξ can have such that it admits a gradient-like vector field without homoclinic cycles and give estimations for this number.

Topological Complexity of Motion Planning

Abstract. In this paper we study a notion of topological complexity TC(X) for the planning problem. TC(X) is a number which measures discontinuity of the process of planning in the configuration space X . More precisely, TC(X) is the minimal number k such that there are k different motion planning rules, each defined on an open subset of X× X , so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik—Schnirelman theory) to study the topological complexity TC(X) . We give an upper bound for TC(X) (in terms of the dimension of the configuration space X ) and also a lower bound (in terms of the structure of the cohomology algebra of X ). We explicitly compute the topological complexity of planning for a number of configuration spaces: spheres, two-dimensional surfaces, products of spheres. In particular, we completely calculate the topological complexity of the problem of planning for a robot arm in the absence of obstacles.