Palais-Smale Condition Research Articles

Homotopy properties of horizontal loop spaces and applications to closed sub-Riemannian geodesics

Given a manifold M M and a proper sub-bundle Δ ⊂ T M \Delta \subset TM , we investigate homotopy properties of the horizontal free loop space Λ \Lambda , i.e., the space of absolutely continuous maps γ : S 1 → M \gamma :S^1\to M whose velocities are constrained to Δ \Delta (for example: legendrian knots in a contact manifold). In the first part of the paper we prove that the base-point map F : Λ → M F:\Lambda \to M (the map associating to every loop its base-point) is a Hurewicz fibration for the W 1 , 2 W^{1,2} topology on Λ \Lambda . Using this result we show that, even if the space Λ \Lambda might have deep singularities (for example: constant loops form a singular manifold homeomorphic to M M ), its homotopy can be controlled nicely. In particular we prove that Λ \Lambda (with the W 1 , 2 W^{1,2} topology) has the homotopy type of a CW-complex, that its inclusion in the standard free loop space (i.e., the space of loops with no non-holonomic constraint) is a homotopy equivalence, and consequently that its homotopy groups can be computed as π k ( Λ ) ≃ π k ( M ) ⋉ π k + 1 ( M ) \pi _k(\Lambda )\simeq \pi _k(M) \ltimes \pi _{k+1}(M) for all k ≥ 0. k\geq 0. In the second part of the paper we address the problem of the existence of closed sub-Riemannian geodesics. In the general case we prove that if ( M , Δ ) (M, \Delta ) is a compact sub-Riemannian manifold, each non-trivial homotopy class in π 1 ( M ) \pi _1(M) can be represented by a closed sub-Riemannian geodesic. In the contact case, we prove a min-max result generalizing the celebrated Lyusternik-Fet theorem: if ( M , Δ ) (M, \Delta ) is a compact, contact manifold, then every sub-Riemannian metric on Δ \Delta carries at least one closed sub-Riemannian geodesic. This result is based on a combination of the above topological results with the delicate study of an analogue of a Palais-Smale condition in the vicinity of abnormal loops (singular points of Λ \Lambda ).

The energy identity for a sequence of Yang–Mills $$\alpha $$-connections

We prove that the Yang–Mills $$\alpha $$ -functional satisfies the Palais–Smale condition, implying the existence of critical points, which are called Yang–Mills $$\alpha $$ -connections. It was shown in Hong et al. (Comment Math Helv 90:75–120, 2015) that as $$\alpha \rightarrow 1$$ , a sequence of Yang–Mills $$\alpha $$ -connections converges to a Yang–Mills connection away from finitely many points. We prove an energy identity for such a sequence of Yang–Mills $$\alpha $$ -connections. As an application, we also prove an energy identity for the Yang–Mills flow at the maximal existence time.

A multiplicity result for a class of fractional p-Laplacian equations with perturbations in ℝN

ABSTRACTThis paper deals with a class of nonlinear elliptic equations with perturbations in the whole space involving the fractional p-Laplacian. As a particular case, we investigate the following Schrödinger equations with perturbations: where is the fractional p-Laplacian operator, is a positive continuous function, is a perturbation. We first establish a compactness theorem which allows us to give some estimates of the energy levels where the Palais-Smale condition can fail. Furthermore, using Ekeland's variational principle and the mountain pass theorem, we obtain the existence of at least two distinct nonnegative weak solutions for the above-mentioned equations.

Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations

In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs (N×N)-system:Δui=λ(∑j=1N∑k=1NKkjKjieujeuk−∑j=1NKjieuj)+4π∑j=1niδpij,i=1,…,N; over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of SU(N+1), (see (1.2) below). Here, λ>0 is the coupling parameter, δp is the Dirac measure with pole at p and ni∈N, for i=1,…,N. When N=1,2 many results are now available for the periodic solvability of such system and provide the existence of different classes of solutions known as: topological, non-topological, mixed and blow-up type. On the contrary for N≥3, only recently in [27] the authors managed to obtain the existence of one doubly periodic solution via a minimization procedure, in the spirit of [46]. Our main contribution in this paper is to show (as in [46]) that actually the given system admits a second doubly periodic solutions of “Mountain-pass” type, provided that 3≤N≤5. Note that the existence of multiple solutions is relevant from the physical point of view. Indeed, it implies the co-existence of different non-Abelian Chern–Simons condensates sharing the same set (assigned component-wise) of vortex points, energy and fluxes. The main difficulty to overcome is to attain a “compactness” property encompassed by the so-called Palais–Smale condition for the corresponding “action” functional, whose validity remains still open for N≥6.

Nonlinear Scalar Field Equations with L 2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches

Abstract We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in ℝ N {\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ): ${(*)_{m}}$ { - Δ ⁢ u = g ⁢ ( u ) - μ ⁢ u in ⁢ ℝ N , ∥ u ∥ L 2 ⁢ ( ℝ N ) 2 = m , u ∈ H 1 ⁢ ( ℝ N ) , \displaystyle\begin{cases}-\Delta u=g(u)-\mu u\quad\text{in }\mathbb{R}^{N},% \cr\lVert u\rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m,\cr u\in H^{1}(\mathbb{R}^{N})% ,\end{cases} where g ⁢ ( ξ ) ∈ C ⁢ ( ℝ , ℝ ) {g(\xi)\in C(\mathbb{R},\mathbb{R})} , m > 0 {m>0} is a given constant and μ ∈ ℝ {\mu\in\mathbb{R}} is a Lagrange multiplier. We introduce a new approach using a Lagrange formulation of problem ( * ) m {(*)_{m}} . We develop a new deformation argument under a new version of the Palais–Smale condition. For a general class of nonlinearities related to [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 313–345], [H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983), no. 4, 347–375], [J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in ℝ N {\mathbb{R}^{N}} : Mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal. 35 (2010), no. 2, 253–276], it enables us to apply minimax argument for L 2 {L^{2}} constraint problems and we show the existence of infinitely many solutions as well as mountain pass characterization of a minimizing solution of the problem inf ⁡ { ∫ ℝ N 1 2 ⁢ | ∇ ⁡ u | 2 - G ⁢ ( u ) ⁢ d ⁢ x : ∥ u ∥ L 2 ⁢ ( ℝ N ) 2 = m } , G ⁢ ( ξ ) = ∫ 0 ξ g ⁢ ( τ ) ⁢ 𝑑 τ . \inf\Bigg{\{}\int_{\mathbb{R}^{N}}{1\over 2}|{\nabla u}|^{2}-G(u)\,dx:\lVert u% \rVert_{L^{2}(\mathbb{R}^{N})}^{2}=m\Bigg{\}},\quad G(\xi)=\int_{0}^{\xi}g(% \tau)\,d\tau.

The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

Nonlinear time-harmonic Maxwell equations in a bounded domain: Lack of compactness

We survey recent results on ground and bound state solutions $E:\Omega\to\mathbb{R}^3$ of the problemon a bounded Lipschitz domain $\Omega\subset\mathbb{R}^3$, where $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$. The equation describes the propagation of the time-harmonic electric field $\Re\{E(x){\rm~e}^{{\rm~i}\omega~t}\}$ in a nonlinear isotropic material $\Omega$ with $\lambda=-\mu~\varepsilon~\omega^2\leq~0$, where $\mu$ and $\varepsilon$ stand forthe permeability and the linear part of the permittivity of the material. The nonlinear term $|E|^{p-2}E$ with $2<p\leq~2^*=6$ comes from the nonlinear polarization andthe boundary conditions are those for $\Omega$ surrounded by a perfect conductor. The problem has a variational structure; however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition. We show the underlying difficulties of the problem and enlist some open questions.

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.

Existence and multiplicity of weak solutions for gradient-type systems wıth oscillatory nonlinearities on the sierpinski gasket

In this paper, we establish the existence and multiplicity results of solutions for parametric quasi-linear systems of the gradient-type on the Sierpinski gasket is proved. Our technical approach is based on variational methods and critical points theory and on certain analytic and geometrical properties of the Sierpinski fractal. Indeed, using a consequence of the local minimum theorem due to Bonanno, the Palais-Smale condition cut off upper at $r$, and the Palais-Smale condition for the Euler functional we investigate the existence of one and two solutions for our problem under algebraic conditions on the nonlinear part. Moreover by applying a different three critical point theorem due to Bonanno and Marano we guarantee the existence of third solution for our problem.

ELLIPTIC BOUNDARY VALUE PROBLEM WITH TWO SINGULARITIES

We investigate existence and multiplicity of the solutions for elliptic boundary value problem with two singularities. We obtain one theorem which shows that there exists at least one nontrivial weak solution under some conditions on which the corresponding functional of the problem satisfies the Palais-Smale condition. We obtain this result by variational method and critical point theory.

Infinitely many solutions for quasilinear elliptic equations with lack of symmetry

In this paper we look for weak solutions of the quasilinear elliptic model problem −div(A(x,u)∇u)+12At(x,u)|∇u|2=g(x,u)+h(x)in Ω,u=0on ∂Ω,where Ω⊂RN is a bounded domain, N≥2, the real terms A(x,t), At(x,t)=∂A∂t(x,t) and g(x,t) are Carathéodory functions on Ω×R and h:Ω→R is a given measurable map.We prove that, even if At(x,t)≢0, under suitable assumptions infinitely many solutions exist in spite of the lack of symmetry. A suitable supercritical growth is allowed for the nonlinear term g(x,t).We use a variant of the variational perturbation techniques introduced by Rabinowitz in Rabinowitz (1982) but by means of a weak version of the Cerami–Palais–Smale condition.

Existence of bounded variation solutions for a 1-Laplacian problem with vanishing potentials

In this work it is studied a quasilinear elliptic problem in the whole space RN involving the 1-Laplacian operator, with potentials which can vanish at infinity. The Euler–Lagrange functional is defined in a space whose definition resembles BV(RN). It is proved the existence of a nonnegative nontrivial bounded variation solution and the proof relies on a version of the Mountain Pass Theorem without the Palais–Smale condition to Lipschitz continuous functionals.

A viscosity method in the min-max theory of minimal surfaces

We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface $\Sigma $ into a given closed manifold, we add to the area Lagrangian a term equal to the $L^{q}$ norm of the second fundamental form of the immersion times a “viscosity” parameter. This relaxation of the area functional satisfies the Palais–Smale condition for $q>2$ . This permits to construct critical points of the relaxed Lagrangian using classical min-max arguments such as the mountain pass lemma. The goal of this work is to describe the passage to the limit when the “viscosity” parameter tends to zero. Under some natural entropy condition, we establish a varifold convergence of these critical points towards a parametrized integer stationary varifold realizing the min-max value. It is proved in Pigati and Riviere ( arXiv:1708.02211 , 2017) that parametrized integer stationary varifold are given by smooth maps exclusively. As a consequence we conclude that every surface area minmax is realized by a smooth possibly branched minimal immersion.

Existence of weak solutions to degenerate p-Laplacian equations and integral formulas

We study the problem of solving some general integral formulas and then apply the conclusions to obtain results about the existence of weak solutions of various degenerate p-Laplacian equations. We adapt Variational Calculus methods and the Mountain Pass Lemma without the Palais–Smale condition, and we use an abstract version of Lions' Concentration Compactness Principle II.

Existence and Asymptotic Behaviour for a Kirchhoff Type Equation With Variable Critical Growth Exponent

In this paper, we establish existence and asymptotic behaviour of nontrivial weak solution of a class of quasilinear stationary Kirchhoff type equations involving the variable exponent spaces with critical growth, namely $$\begin{aligned}{\left\{ \begin{array}{ll} -M (\mathcal{A}(u)) {\rm div} (a(|\nabla u|^{p(x)}) | \nabla u|^{p(x) - 2} \nabla u) = \lambda f (x, u) + |u|^{s(x)-2} u \quad {\rm in} \quad \Omega,\\ u = 0 \quad {\rm on} \quad \partial \Omega,\end{array}\right. } \end{aligned}$$ where \({\Omega}\) is a bounded smooth domain of \({\mathbb{R}^N}\) , with homogeneous Dirichlet boundary conditions on \({\partial \Omega}\) , the nonlinearities \({f : \Omega \times \mathbb{R} \rightarrow \mathbb{R}}\) is a continuous function, \({a : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}}\) is a function of the class \({C^1}\) , \({M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}}\) is a continuous function whose properties will be introduced later, and \({\lambda}\) is a positive parameter. We assume that \({\mathcal{C} = \{x \in \Omega : s(x) = \gamma^{*}(x)\} \neq \emptyset}\) , where \({\gamma (x)^{*} = N \gamma (x) / (N - \gamma (x))}\) is the critical Sobolev exponent. We show that the problem has at least one solution, which it converges to zero, in the norm of the space X as \({\lambda \rightarrow + \infty}\) . Our result extends, complement and complete in several ways some of the recent works. We want to emphasize that a difference of some previous research is that the conditions on \({a(\cdot)}\) are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for \({p(x) > 1}\) , for all \({x \in \bar{\Omega}}\) . The main tools used are the Mountain Pass Theorem without the Palais-Smale condition given in [11] and the Concentration Compactness Principle for variable exponent found in [9]. We remark that it will be necessary a suitable truncation argument in the Euler- Lagrange operator associated.

Triple solutions for quasilinear one-dimensional $p$ -Laplacian elliptic equations in the whole space

In this paper, we establish the existence of three possibly nontrivial solutions for a Dirichlet problem on the real line without assuming on the nonlinearity asymptotic conditions at infinity. As a particular case, when the nonlinearity is superlinear at zero and sublinear at infinity, the existence of two nontrivial solutions is obtained. This approach is based on variational methods and, more precisely, a critical points theorem, which assumes a more general condition than the classical Palais–Smale condition, is exploited.

Variational methods for non-variational problems

We propose and describe an alternative perspective for the study of systems of boundary value problems governed by ODEs. It is based on a variational approach that seeks to minimize a certain quadratic error understood as a deviation of paths from being a solution of the corresponding system. We distinguish two situations depending on whether the problem has or has not divergence structure. In the first case, the functional is not a typical integral functional as the ones examined in the Calculus of Variations, and we have to resort to the Palais–Smale condition to show existence of minimizers. In the case of a fully non-linear problem, however, the functional is an integral, local functional of second order. We illustrate the method with some numerical simulations.

Morse theory for elastica

In Riemannian manifolds the elastica are critical points of the restriction of total squared geodesic curvature to curves with fixed length which satisfy first order boundary conditions. We verify that the Palais-Smale condition holds for this variational problem, and also the related problems where the admissible curves are required to satisfy zeroth order boundary conditions, or first order periodicity conditions. We also prove a Morse index theorem for elastica and use the Morse inequalities to give lower bounds for the number of elastica of each index in terms of the Betti numbers of the path space.

Singular potential biharmonic problem with fixed energy

We investigate multiple solutions for the perturbation of a singular potential biharmonic problem with fixed energy. We get a theorem that shows the existence of at least one nontrivial weak solution under some conditions and fixed energy on which the corresponding functional of the equation satisfies the Palais-Smale condition. We obtain this result by variational method and critical point theory.

Degenerate nonlinear elliptic equations lacking in compactness

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form −div(a(x)∇pu(x)) = g(λ, x, |u|p−2u) in ℝN, where ∇pu = |∇u|p−2∇u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.