Mostrando recursos 1 - 20 de 75

  1. Geodesics on ${\rm SO}(n)$ and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

    Day, Stuart; Taheri, Ali
    We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system $$ \begin{cases} -{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u & \text{in } \mathbb{X}^n,\\ |u| = 1 &\text{in } \mathbb{X}^n ,\\ u = \varphi &\text{on } \partial \mathbb{X}^n. \end{cases} $$ Here $\varphi \in \mathscr{C}^\infty(\partial \mathbb X^n, \mathbb S^{n-1})$ is a suitable boundary map, $F'$ is the derivative of $F$ with respect to the second argument, $u \in W^{1,p}(\mathbb X^n, \mathbb S^{n-1})$ for a fixed $1

  2. On Doubly Nonlocal $p$-fractional Coupled Elliptic System

    Mukherjee, Tuhina; Sreenadh, Konijeti
    We study the following nonlinear system with perturbations involving $p$-fractional Laplacian: \begin{equation} \begin{cases} (-\Delta)^s_p u+ a_1(x)u|u|^{p-2} = \alpha(|x|^{-\mu}*|u|^q)|u|^{q-2}u\\ \hskip 2.5 cm + \beta (|x|^{-\mu}*|v|^q)|u|^{q-2}u+ f_1(x) & \text{in } \mathbb R^n, \\ (-\Delta)^s_p v+ a_2(x)v|v|^{p-2} = \gamma(|x|^{-\mu}*|v|^q)|v|^{q-2}v \\ \hskip 2.5cm + \beta (|x|^{-\mu}*|u|^q)|v|^{q-2}v+ f_2(x)& \text{in } \mathbb R^n, \end{cases} \tag{P} \end{equation} where $n>sp$, $0< s< 1$, $p\geq2$, $\mu \in (0,n)$, ${p}( 2-{\mu}/{n})/2 < q <{p^*_s}( 2-{\mu}/{n})/2$, $\alpha,\beta,\gamma >0$, $0< a_i \in C(\mathbb R^n, \mathbb R)$, $i=1,2$ and $f_1,f_2\colon \mathbb R^n \to \mathbb R$ are perturbations. We show existence of at least two nontrivial solutions for (P) using Nehari manifold and minimax...

  3. A note on dimensional entropy for amenable group actions

    Dou, Dou; Zhang, Ruifeng
    In this short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along tempered Følner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some Følner sequence) equals its topological entropy. This answers questions by Zheng and Chen [10] and Simpson [9].

  4. A gradient flow generated by a nonlocal model of a neural field in an unbounded domain

    da Silva, Severino Horacio; Pereira, Antônio Luiz
    In this paper we consider the nonlocal evolution equation \[ \frac{\partial u(x,t)}{\partial t} + u(x,t)= \int_{\mathbb{R}^{N}}J(x-y)f(u(y,t))\rho(y)\,dy+ h(x). \] We show that this equation defines a continuous flow in both the space $C_{b}(\mathbb{R}^{N})$ of bounded continuous functions and the space $C_{\rho}(\mathbb{R}^{N})$ of continuous functions $u$ such that $u \cdot \rho$ is bounded, where $\rho $ is a convenient ``weight function''. We show the existence of an absorbing ball for the flow in $C_{b}(\mathbb{R}^{N})$ and the existence of a global compact attractor for the flow in $C_{\rho}(\mathbb{R}^{N})$, under additional conditions on the nonlinearity. We then exhibit a continuous Lyapunov function which is...

  5. On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian

    Audoux, Benjamin; Bobkov, Vladimir; Parini, Enea
    We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\Omega \subset \mathbb{R}^N$. By means of topological arguments, we show how symmetries of $\Omega$ help to construct subsets of $W_0^{1,p}(\Omega)$ with suitably high Krasnosel'skiĭ genus. In particular, if $\Omega$ is a ball $B \subset \mathbb{R}^N$, we obtain the following chain of inequalities: \[ \lambda_2(p;B) \leq \dots \leq \lambda_{N+1}(p;B) \leq \lambda_\ominus(p;B). \] Here $\lambda_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $\lambda_\ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial...

  6. Dynamics on sensitive and equicontinuous functions

    Li, Jie; Yu, Tao; Zeng, Tiaoying
    The notions of sensitive and equicontinuous functions under semigroup action are introduced and intensively studied. We show that a transitive system is sensitive if and only if it has a sensitive pair if and only if it has a sensitive function. While there exists a minimal non-weakly mixing system such that every non-constant continuous function is sensitive, and a topological dynamical system is weakly mixing if and only if it is sensitive consistently with respect to (at least) any two non-constant continuous functions. We also get a dichotomy result for minimal systems - every continuous function is either sensitive or...

  7. Sign changing solutions of $p$-fractional equations with concave-convex nonlinearities

    Bhakta, Mousomi; Mukherjee, Debangana
    We study the existence of sign changing solutions to the following $p$-fractional problem with concave-critical nonlinearities: \begin{alignat*}2 (-\Delta)^s_pu &= \mu |u|^{q-1}u + |u|^{p^*_s-2}u &\quad&\mbox{in }\Omega,\\ u&=0&\quad&\mbox{in } \mathbb{R}^N\setminus\Omega, \end{alignat*} where $s\in(0,1)$ and $p\geq 2$ are fixed parameters, $0< q< p-1$, $\mu\in\mathbb{R}^+$ and $p_s^*={Np}/({N-ps})$. $\Omega$ is an open, bounded domain in $\mathbb{R}^N$ with smooth boundary, $N> ps$.

  8. A version of Krasnosel'skiĭ's compression-expansion fixed point theorem in cones for discontinuous operators with applications

    Figueroa, Rubén; Pouso, Rodrigo López; Rodríguez-López, Jorge
    We introduce a new fixed point theorem of Krasnosel'skiĭ type for discontinuous operators. As an application we use it to study the existence of positive solutions of a second-order differential problem with separated boundary conditions and discontinuous nonlinearities.

  9. On a singular semilinear elliptic problem: multiple solutions via critical point theory

    Faraci, Francesca; Smyrlis, George
    We study existence and multiplicity of solutions of a semilinear elliptic problem involving a singular term. Combining various techniques from critical point theory, under different sets of assumptions, we prove the existence of $k$ solutions ($k\in\mathbb N$) or infinitely many weak solutions.

  10. Nonlinear Unilateral Parabolic Problems in Musielak--Orlicz spaces with $L^1$ data

    Ait Khellou, Mustafa; Douiri, Sidi Mohamed; El Hadfi, Youssef
    We study, in Musielak-Orlicz spaces, the existence of solutions for some strongly nonlinear parabolic unilateral problem with $L^1$ data and without sign condition on nonlinearity.

  11. General and optimal decay for a viscoelastic equation with boundary feedback

    Messaoudi, Salim A.; Al-Khulaifi, Waled
    We establish a general decay rate for a viscoelastic problem with a nonlinear boundary feedback and a relaxation function satisfying $g'(t) \leq - \xi(t) g^{p}(t)$, $t\geq 0$, $ 1\leq p < 3/2$. This work generalizes and improves earlier results in the literature. In particular those of [5], [11] and [17].

  12. Existence of solutions for a class of degenerate elliptic equations in $P(x)$-Sobolev spaces

    Aharrouch, Benali; Boukhrij, Mohamed; Bennouna, Jouad
    We study the Dirichlet problem for degenerate elliptic equations of the form \begin{equation*} - \mbox{div} \, a(x,u,\nabla u)+ H(x,u,\nabla u)= f \quad \mbox{in } \Omega, \end{equation*} where $ a(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown $u$, and $H(x,u,\nabla u)$ is a nonlinear term without sign condition. Under suitable conditions on $a$ and $H$, we prove the existence of bounded and unbounded solution for a datum $f\in L^m$, with $1\leq m\leq \infty$.

  13. Existence of a weak solution for the fractional $p$-Laplacian equations with discontinuous nonlinearities via the Berkovits-Tienari degree theory

    Kim, Yun-Ho
    We are concerned with the following nonlinear elliptic equations of the fractional $p$-Laplace type: \begin{equation*} \begin{cases} (-\Delta)_p^su \in \lambda[\underline{f}(x,u(x)), \overline{f}(x,u(x))] &\textmd{in } \Omega,\\ u= 0 &\text{on } \mathbb{R}^N\setminus\Omega, \end{cases} \end{equation*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $\lambda$ is a parameter, $0 \lt s \lt 1\lt p\lt +\infty$, $sp \lt N$, and the measurable functions $\underline{f}$, $ \overline{f}$ are induced by a possibly discontinuous at the second variable function $f\colon \Omega\times\mathbb R \to \mathbb R$. By using the Berkovits-Tienari degree theory for upper semicontinuous set-valued operators of type (S$_+)$ in reflexive Banach spaces, we show that our problem with the...

  14. A class of De Giorgi type and local boundedness

    Liu, Duchao; Yao, Jinghua
    Under appropriate assumptions on the $N(\Omega)$-function, the De Giorgi process is presented in the framework of Musielak-Orlicz-Sobolev spaces. As the applications, the local boundedness property of the minimizers for a class of the energy functionals in Musielak-Orlicz-Sobolev spaces is proved; and furthermore, the local boundedness of the weak solutions for a class of fully nonlinear elliptic equations is provided.

  15. Existence of three nontrivial solutions for a class of fourth-order elliptic equations

    Li, Chun; Agarwal, Ravi P.; Ou, Zeng-Qi
    The existence of three nontrivial solutions is established for a class of fourth-order elliptic equations. Our technical approach is based on Linking Theorem and ($\nabla$)-Theorem.

  16. 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.

  17. 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...

  18. 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].

  19. 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.

  20. 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.

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