Mostrando recursos 1 - 20 de 32

  1. Critical Brezis-Nirenberg problem for nonlocal systems

    Faria, Luiz F.O.; Miyagaki, Olimpio H.; Pereira, Fábio R.
    We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.

  2. Critical Brezis-Nirenberg problem for nonlocal systems

    Faria, Luiz F.O.; Miyagaki, Olimpio H.; Pereira, Fábio R.
    We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.

  3. Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms

    Teng, Kaimin; Agarwal, Ravi P.
    We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

  4. Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms

    Teng, Kaimin; Agarwal, Ravi P.
    We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

  5. A noncommutative version of Farber's topological complexity

    Manuilov, Vladimir
    Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative $C^*$-algebras.

  6. A noncommutative version of Farber's topological complexity

    Manuilov, Vladimir
    Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative $C^*$-algebras.

  7. Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms

    Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca
    We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term is sign changing, then again we produce a sequence of smooth solutions converging to zero in $C^1(\overline{\Omega})$, but we cannot claim that they are nodal.

  8. Multiple nodal solutions for semilinear Robin problems with indefinite linear part and concave terms

    Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca
    We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term is sign changing, then again we produce a sequence of smooth solutions converging to zero in $C^1(\overline{\Omega})$, but we cannot claim that they are nodal.

  9. Periodic solutions of vdP and vdP-like systems on 3-tori

    Balanov, Zalman; Hooton, Edward; Murza, Adrian
    Van der Pol equation (in short, vdP) as well as many its non-symmetric generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled networks modeling real-life phenomena. Symmetric properties of periodic regimes of networks of vdP/vdPl depend on symmetries of coupling. In this paper, we consider $N^3$ identical vdP/vdPl oscillators arranged in a cubical lattice, where opposite faces are identified in the same way as for a $3$-torus. Depending on which nodes impact the dynamics of a given node, we distinguish between $\mathbb D_N \times \mathbb D_N \times \mathbb D_N$-equivariant systems and their $\mathbb Z_N \times \mathbb Z_N \times \mathbb Z_N$-equivariant counterparts....

  10. Periodic solutions of vdP and vdP-like systems on 3-tori

    Balanov, Zalman; Hooton, Edward; Murza, Adrian
    Van der Pol equation (in short, vdP) as well as many its non-symmetric generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled networks modeling real-life phenomena. Symmetric properties of periodic regimes of networks of vdP/vdPl depend on symmetries of coupling. In this paper, we consider $N^3$ identical vdP/vdPl oscillators arranged in a cubical lattice, where opposite faces are identified in the same way as for a $3$-torus. Depending on which nodes impact the dynamics of a given node, we distinguish between $\mathbb D_N \times \mathbb D_N \times \mathbb D_N$-equivariant systems and their $\mathbb Z_N \times \mathbb Z_N \times \mathbb Z_N$-equivariant counterparts....

  11. Multiplicity of positive solutions for Kirchhoff type problems in $\mathbb R^3$

    Hu, Tingxi; Lu, Lu
    We are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle - \bigg(\varepsilon ^2a + \varepsilon b\int_{{\mathbb{R}^3}}{|\nabla u{|^2}} dx \bigg)\Delta u + u = Q(x)|u|^{p-2}u,& x\in \mathbb{R}^3, \hfill \\ u \in H^1(\mathbb{R}^3), \quad u > 0, & x\in \mathbb{R}^3 , \end{cases} \end{equation*} where $\varepsilon> 0 $ is a parameter, $a, b> 0$ are constants, $p\in (2, 6)$, and $Q\in C(\mathbb{R}^3)$ is a nonnegative function. We show how the profile of $Q$ affects the number of positive solutions when $\varepsilon $ is sufficiently small.

  12. Multiplicity of positive solutions for Kirchhoff type problems in $\mathbb R^3$

    Hu, Tingxi; Lu, Lu
    We are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle - \bigg(\varepsilon ^2a + \varepsilon b\int_{{\mathbb{R}^3}}{|\nabla u{|^2}} dx \bigg)\Delta u + u = Q(x)|u|^{p-2}u,& x\in \mathbb{R}^3, \hfill \\ u \in H^1(\mathbb{R}^3), \quad u > 0, & x\in \mathbb{R}^3 , \end{cases} \end{equation*} where $\varepsilon> 0 $ is a parameter, $a, b> 0$ are constants, $p\in (2, 6)$, and $Q\in C(\mathbb{R}^3)$ is a nonnegative function. We show how the profile of $Q$ affects the number of positive solutions when $\varepsilon $ is sufficiently small.

  13. Generalized recurrence in impulsive semidynamical systems

    Ding, Boyang; Ding, Changming
    We aim to introduce the generalized recurrence into the theory of impulsive semidynamical systems. Similarly to Auslander's construction in [J. Auslander, Generalized recurrence in dynamical systems, Contrib. Differential Equations 3 (1964), 65-74], we present two different characterizations, respectively, by Lyapunov functions and higher prolongations. In fact, we show that if the phase space is a locally compact separable metric space, then the generalized recurrent set is the same as the quasi prolongational recurrent set. Also, we see that many new phenomena appear for the impulse effects in the semidynamical system.

  14. Generalized recurrence in impulsive semidynamical systems

    Ding, Boyang; Ding, Changming
    We aim to introduce the generalized recurrence into the theory of impulsive semidynamical systems. Similarly to Auslander's construction in [J. Auslander, Generalized recurrence in dynamical systems, Contrib. Differential Equations 3 (1964), 65-74], we present two different characterizations, respectively, by Lyapunov functions and higher prolongations. In fact, we show that if the phase space is a locally compact separable metric space, then the generalized recurrent set is the same as the quasi prolongational recurrent set. Also, we see that many new phenomena appear for the impulse effects in the semidynamical system.

  15. Coincidence degree methods in almost periodic differential equations

    Qi, Liangping; Yuan, Rong
    We consider the existence of almost periodic solutions to differential equations by using coincidence degree theory. A new equivalent spectral condition for the compactness of integral operators on almost periodic function spaces is established. It is shown that semigroup conditions are crucial in applications.

  16. Coincidence degree methods in almost periodic differential equations

    Qi, Liangping; Yuan, Rong
    We consider the existence of almost periodic solutions to differential equations by using coincidence degree theory. A new equivalent spectral condition for the compactness of integral operators on almost periodic function spaces is established. It is shown that semigroup conditions are crucial in applications.

  17. On the dynamics of a modified Cahn-Hilliard equation with biological applications

    Zhao, Xiaopeng
    We study the global solvability and dynamical behaviour of the modified Cahn-Hilliard equation with biological applications in the Sobolev space $H^1(\mathbb{R}^N)$.

  18. On the dynamics of a modified Cahn-Hilliard equation with biological applications

    Zhao, Xiaopeng
    We study the global solvability and dynamical behaviour of the modified Cahn-Hilliard equation with biological applications in the Sobolev space $H^1(\mathbb{R}^N)$.

  19. Existence of solutions for nonlinear $p$-Laplacian difference equations

    Saavedra, Lorena; Tersian, Stepan
    The aim of this paper is the study of existence of solutions for nonlinear $2n^{\rm th}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.

  20. Existence of solutions for nonlinear $p$-Laplacian difference equations

    Saavedra, Lorena; Tersian, Stepan
    The aim of this paper is the study of existence of solutions for nonlinear $2n^{\rm th}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.

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