Mostrando recursos 1 - 20 de 60

  1. On the topological pressure of the saturated set with non-uniform structure

    Zhao, Cao; Chen, Ercai
    We derive a conditional variational principle of the saturated set for systems with the non-uniform structure. Our result applies to a broad class of systems including $\beta$-shifts, $S$-gap shifts and their subshift factors.

  2. Cancellations for circle-valued Morse functions via spectral sequences

    Lima, Dahisy V. de S.; Neto, Oziride Manzoli; de Rezende, Ketty A.; da Silveira, Mariana R.
    A spectral sequence analysis of a filtered Novikov complex $(\mathcal{N}_{\ast}(f),\Delta)$ over $\mathbb{Z}((t))$ is developed with the goal of obtaining results relating the algebraic and dynamical settings. Specifically, the unfolding of a spectral sequence of $(\mathcal{N}_{\ast}(f),\Delta)$ and the cancellation of its modules is associated to a one parameter family of circle-valued Morse functions on a~surface and the dynamical cancellations of its critical points. The data of a spectral sequence computed for $(\mathcal{N}_{\ast}(f),\Delta)$ is encoded in a family of matrices $\Delta^r$ produced by the Spectral Sequence Sweeping Algorithm (SSSA), which has as its initial input the differential $\Delta$. As one ``turns the...

  3. Poisson structures on closed manifolds

    Mukherjee, Sauvik
    We prove an $h$-principle for Poisson structures on closed manifolds. Equivalently, we prove an $h$-principle for symplectic foliations (singular) on closed manifolds. On open manifolds however the singularities could be avoided and it is a known result by Fernandes and Frejlich [Int. Math. Res. Not. IMRN (2012), no. 7, 1505-1518].

  4. Ground state solutions for a class of semilinear elliptic systems with sum of periodic and vanishing potentials

    Che, Guofeng F.; Chen, Haibo B.
    In this paper, we consider the following semilinear elliptic systems: $$ \begin{cases} -\Delta u+V(x)u=F_{u}(x, u, v)-\Gamma(x)|u|^{q-2}u & \mbox{in }\mathbb{R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v)-\Gamma(x)|v|^{q-2}v & \mbox{in }\mathbb{R}^{N},\\ \end{cases} $$ where $q\in[2,2^{*})$, $V=V_{{\rm per}}+V_{{\rm loc}}\in L^{\infty}(\mathbb{R}^{N})$ is the sum of a periodic potential $V_{{\rm per}}$ and a localized potential $V_{{\rm loc}}$ and $\Gamma\in L^{\infty}(\mathbb{R}^{N})$ is periodic and $\Gamma(x)\geq0$ for almost every $x\in\mathbb{R}^{N}$. Under some appropriate assumptions on $F$, we investigate the existence and nonexistence of ground state solutions for the above system. Recent results from the literature are improved and extended.

  5. Singular levels and topological invariants of Morse-Bott foliations on non-orientable surfaces

    Martínez-Alfaro, José; Meza-Sarmiento, Ingrid S.; Oliveira, Regilene D.S.
    We investigate the classification of closed curves and eight curves of saddle points defined on non-orientable closed surfaces, up to an ambient homeomorphism. The classification obtained here is applied to Morse-Bott foliations on non-orientable closed surfaces in order to define a complete topological invariant.

  6. Hardy-Sobolev inequality with singularity a curve

    Fall, Mouhamed Moustapha; Thiam, El hadji Abdoulaye
    We consider a bounded domain $\Omega$ of $\mathbb{R}^N$, $N\ge3$, and $h$ a continuous function on $\Omega$. Let $\Gamma$ be a closed curve contained in $\Omega$. We study existence of positive solutions $u \in H^1_0(\Omega)$ to the equation \[ -\Delta u+h u=\rho^{-\sigma}_\Gamma u^{2^*_\sigma-1} \quad \textrm{in } \Omega, \] where $2^*_\sigma:={2(N-\sigma)}/({N-2})$, $\sigma\in (0,2)$, and $\rho_\Gamma$ is the distance function to $\Gamma$. For $N\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of...

  7. On the well-posedness of differential mixed quasi-variational-inequalities

    Liu, Zhenhai; Motreanu, Dumitru; Zeng, Shengda
    We discuss the well-posedness and the well-posedness in the generalized sense of differential mixed quasi-variational inequalities ((DMQVIs), for short) in Hilbert spaces. This gives us an outlook to the convergence analysis of approximating sequences of solutions for (DMQVIs). Using these concepts we point out the relation between metric characterizations and well-posedness of (DMQVIs). We also prove that the solution set of (DMQVIs) is compact, if problem (DMQVIs) is well-posed in the generalized sense.

  8. Existence and concentration of ground state sign-changing solutions for Kirchhoff type equations with steep potential well and nonlinearity

    Chen, Jianhua; Tang, Xianhua; Cheng, Bitao
    We study the following class of elliptic equations: \begin{equation*} -\bigg(a+b\int_{{\mathbb R}^3}|\nabla u|^2\,dx\bigg)\Delta u+\lambda V(x)u=f(u), \quad x\in{\mathbb R}^3, \end{equation*} where $\lambda,a,b>0$, $V\in \mathcal{C}({\mathbb R}^3,{\mathbb R})$ and $V^{-1}(0)$ has nonempty interior. First, we obtain one ground state sign-changing solution $u_{b,\lambda}$ applying the non-Nehari manifold method. We show that the energy of $u_{b,\lambda}$ is strictly larger than twice that of the ground state solutions of Nehari-type. Next we establish the convergence property of $u_{b,\lambda}$ as $b\searrow0$. Finally, the concentration of $u_{b,\lambda}$ is explored on the set $V^{-1}(0)$ as $\lambda\rightarrow\infty$.

  9. Dynamics of the BBM equation with a distribution force in low regularity spaces

    Wang, Ming; Liu, Anping
    The Benjamin-Bona-Mahony equation with a distribution force on torus is studied in low regularity spaces. The global well-posedness and the existence of a global attractor in $\dot{H}^{s,p}(\mathbb{T})$ are proved.

  10. Schrödinger-Poisson systems with radial potentials and discontinuous quasilinear nonlinearity

    Mao, Anmin; Chang, Hejie
    We consider the following Schrödinger-Poisson system: $$ \begin{cases} -\triangle u+V(|x|)u+\phi u= Q(|x|) f(u) &\hbox{in } \mathbb{R}^3,\\ -\triangle \phi=u^{2} & \hbox{in } \mathbb{R}^3, \end{cases} $$ with more general radial potentials $V,Q$ and discontinuous nonlinearity $f$. The Lagrange functional may be locally Lipschitz. Using nonsmooth critical point theorem, we obtain the multiplicity results of radial solutions, we also show concentration properties of the solutions. This is in contrast with some recent papers concerning similar problems by using the classical Sobolev embedding theorems.

  11. Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

    Li, Gang; Rădulescu, Vicenţiu D.; Repovš, Dušan D.; Zhang, Qihu
    We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: \begin{equation*} \begin{cases} -{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) &\text{ in }\Omega , \\ u=0 &\text{ on }\partial \Omega . \end{cases} \end{equation*} We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. The present paper extend previous results of Q. Zhang and C. Zhao (Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Computers...

  12. Linear quadratic game of exploitation of common renewable resources with inherent constraints

    Singh, Rajani; Wiszniewska-Matyszkiel, Agnieszka
    In this paper, we analyse a linear quadratic multistage game of extraction of a common renewable resource --- a fishery --- by many players with inherent state dependent constraints for exploitation and an infinite time horizon. To the best of our knowledge, such games have never been studied. We analyse the social optimum and Nash equilibrium for the feedback information structure and compare the results obtained in both cases. For the Nash equilibria, we obtain a value function that is contrary to intuitions from standard linear quadratic games. In our game, we face a situation in which the social optimum...

  13. A class of delay evolution hemivariational inequalities and optimal feedback controls

    Lu, Liang; Liu, Zhenhai; Zhao, Jing
    In this paper, we study the feedback optimal control for a class of evolution hemivariational inequalities with delay. First, we obtain the existence of feasible pairs by applying the Cesari property, the Filippov theorem, the properties of Clarke subdifferential and a fixed point theorem for multivalued maps. Next, the results of optimal feedback control pairs and time optimal control for delay evolution hemivariational inequalities are presented under sufficient conditions. Finally, an example is included to illustrate our main results.

  14. Topological shadowing and the Grobman-Hartman theorem

    Zgliczyński, Piotr
    We give geometric proofs for the Grobman-Hartman theorem for diffeomorphisms and ODEs. Proofs use covering relations and cone conditions for maps and isolating segments and cone conditions for ODEs. We establish topological versions of the Grobman-Hartman theorem as the existence of some semiconjugaces.

  15. A note on Conley index and some parabolic problems with locally large diffusion

    Carbinatto, Maria C.; Rybakowski, Krzysztof P.
    We prove singular Conley index continuation results for a class of scalar parabolic equations with locally large diffusion considered by Fusco [On the explicit construction of an ODE which has the same dynamics as scalar parabolic PDE, J. Differential Equations 69 (1987), 85-110] and Carvalho and Pereira [A scalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations, J. Differential Equations 112 (1994), 81-130].

  16. On a class of cocycles having attractors which consist of singletons

    Guzik, Grzegorz
    We give a new simple sufficient condition for existence of the global pullback attractor which consists of singletons for general cocycle mappings on an arbitrary complete metric space. In particular, we need not have any structure on a parameter space, so the criterion can be applied in both cases: nonautonomous as well as random dynamical systems. Our considerations lead us also to new large class of iterated function systems with point-fibred attractors.

  17. An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators

    Feltrin, Guglielmo; Zanolin, Fabio
    Using Mawhin's coincidence degree theory, we obtain some new continuation theorems which are designed to have as a natural application the study of the periodic problem for cyclic feedback type systems. We also discuss some examples of vector ordinary differential equations with a $\phi$-Laplacian operator where our results can be applied. Our main contribution in this direction is to obtain a continuation theorem for the periodic problem associated with $(\phi(u'))' + \lambda k(t,u,u') = 0$, under the only assumption that $\phi$ is a homeomorphism.

  18. Properties of unique positive solution for a class of nonlocal semilinear elliptic equation

    Jiang, Ruiting; Zhai, Chengbo
    We study a class of nonlocal elliptic equations $$ -M\bigg(\int_{\Omega}|u|^{\gamma}dx\bigg)\Delta u=\lambda f(x,u) $$ with the Dirichlet boundary conditions in bounded domain. Under suitable assumptions on $M$ and the nonlinear term $f$, the existence and new properties of a unique positive solutions are obtained via a monotone operator method and a mixed monotone operator method.

  19. Mayer-Vietoris property of the fixed point index

    Barge, Héctor; Wójcik, Klaudiusz
    We study a Mayer-Vietoris kind formula for the fixed point index of maps of ENR triplets $f\colon (X;X_1,X_2)\to (X;X_1,X_2) $ having compact fixed point set. We prove it under some suitable conditions. For instance when $(X;X_1,X_2)=(E^n;E^n_+,E^n_-)$. We use these results to generalize the Poincaré-Bendixson index formula for vector fields to continuous maps having a sectorial decomposition, to study the fixed point index $i(f,0)$ of orientation preserving homeomorphisms of $E^2_+$ and $(E^3;E^3_+,E^3_-)$ and the fixed point index in the invariant subspace.

  20. Concentration of ground state solutions for fractional Hamiltonian systems

    Torres, César; Zhang, Ziheng
    We are concerned with the existence of ground states solutions to the following fractional Hamiltonian systems: \begin{equation} \begin{cases} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\ u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{cases} \tag*{$(\mbox{FHS})_\lambda$} \end{equation} where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda> 0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies the Ambrosetti-Rabinowitz condition and some other reasonable...

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