We study a $(p,q)$-Laplacian system of Kirchhoff type equations with weight and nonlinearity involving a $(p,q)$-superlinear term, in which $p$ may be different from $q$, and with critical Caffarelli-Kohn-Nirenberg exponent. Using the Mountain Pass Theorem, we obtain a nontrivial solution to the problem.

It is a paper about models for isolated sets and the construction of special hyperbolic Lyapunov functions. We prove that after a suitable surgery every isolated set is the intersection of an attractor and a repeller. We give linear models for attractors and repellers. With these tools we construct hyperbolic Lyapunov functions and metrics around an isolated set whose values along the orbits are catenary curves. Applications are given to expansive flows and homeomorphisms, obtaining, among other things, a hyperbolic metric on local cross sections for an arbitrary expansive flow on a compact metric space.

We study a semilinear Robin problem with an indefinite and unbounded potential and a reaction term which asymptotically at $\pm \infty $ is resonant with respect to any nonprincipal nonnegative eigenvalue. We prove two multiplicity theorems producing three and four nontrivial solutions respectively. Our approach uses variational methods based on the critical point theory, truncation and perturbation techniques, and Morse theory (critical groups).

By using variational methods, we investigate the existence of $T$-periodic solutions to \begin{equation*} \begin{cases} [(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=f(x,u) &\mbox{in } (0,T)^{N}, \\ u(x+Te_{i})=u(x) &\mbox{for all } x \in \mathbb{R}^N, \ i=1, \dots, N, \end{cases} \end{equation*} where $s\in (0,1)$, $N> 2s$, $T> 0$, $m\geq 0$ and $f$ is a continuous function, $T$-periodic in the first variable, verifying the Ambrosetti-Rabinowitz condition, with a polynomial growth at rate $p\in (1, ({N+2s})/({N-2s}))$.

Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a necessary condition to get the deterministic chaos game. As a consequence, we obtain that an IFS of homeomorphisms of the circle is renderable by the deterministic chaos game if and only if it is forward and backward minimal. This result provides examples of attractors (a forward but no backward minimal IFS on the circle) that are not renderable by the deterministic chaos game. We also prove that every well-fibred quasi-attractor is renderable by the deterministic chaos game as well as quasi-attractors of both, symmetric and non-expansive IFSs.

Recursive properties on impulsive semidynamical systems are considered. We obtain results about almost periodic motions and asymptotically almost periodic motions in the context of impulsive systems. The concept of asymptotic almost periodic motions is introduced via time reparametrizations. We also present asymptotic properties for impulsive systems and for their associated discrete systems.

We study the existence of positive ground state solutions for the nonlinear Kirchhoff type equation $$ \begin{cases} \displaystyle -\bigg(a+b\int_{\mathbb R^3}|\nabla u|^2\bigg)\Delta {u}+V(x)u =f(u) & \mbox{in }\mathbb R^3, \\ u\in H^1(\mathbb R^3), \quad u> 0 & \mbox{in } \mathbb R^3, \end{cases} $$ where $a,b> 0$ are constants, $f\in C(\mathbb R,\mathbb R)$ has general critical growth. We generalize a Berestycki-Lions theorem about the critical case of Schrödinger equation to Kirchhoff type equation via variational methods. Moreover, some subcritical works on Kirchhoff type equation are extended to the current critical case.

Many properties or formulas related to the ordinary Euler characteristics of topological spaces are well developed under many mathematical operands, e.g. the product property, fibration property, homotopy axiom, wedge sum property, inclusion-exclusion principle [E.H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966], etc. Unlike these properties, the digital version of the Euler characteristic has its own feature. Among the above properties, we prove that the digital version of the Euler characteristic has the wedge sum property which is of the same type as that for the ordinary Euler characteristic. This property plays an important role in fixed point theory for digital images, digital homotopy theory, digital geometry and so forth.

We are concerned with the following equation: $$ -\varepsilon^2\Delta u+V(x)u=f(u),\quad u(x)> 0\quad \mbox{in } \mathbb{R}^2. $$ By a variational approach, we construct a solution $u_\varepsilon$ which concentrates, as $\varepsilon \to 0$, around arbitrarily given isolated local minima of the confining potential $V$: here the nonlinearity $f$ has a quite general Moser's critical growth, as in particular we do not require the monotonicity of $f(s)/s$ nor the Ambrosetti-Rabinowitz condition.

Building on the construction of least energy sign-changing solutions to variational semilinear elliptic boundary value problems introduced in [A. Castro, J. Cossio and J.M. Neuberger, Sign changing solutions for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), 1041--1053], we prove the existence of a solution with augmented Morse index at least three when a sublevel of the corresponding action functional has nontrivial topology. We provide examples where the set of least energy sign changing solutions is disconnected, hence has nontrivial topology.

In this paper, we study the following Hamiltonian elliptic system with gradient term and critical growth: \begin{equation*} \begin{cases} -\epsilon^{2}\Delta \psi +\epsilon b\cdot \nabla \psi +\psi=K(x)f(|\eta|)\varphi+W(x)|\eta|^{2^*-2}\varphi &\text{in } \mathbb{R}^{N},\\ -\epsilon^{2}\Delta \varphi -\epsilon b\cdot \nabla \varphi +\varphi=K(x)f(|\eta|)\psi+W(x)|\eta|^{2^*-2}\psi &\text{in } \mathbb{R}^{N}, \end{cases} \end{equation*} where $\eta=(\psi,\varphi)\colon \mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $K, W\in C(\mathbb{R}^{N}, \mathbb{R})$, $\epsilon$ is a small positive parameter and $b$ is a constant vector. We require that the nonlinear potentials $K$ and $W$ have at least one global maximum. Combining this with other suitable assumptions on $f$, we prove the existence, exponential decay and concentration phenomena of semiclassical ground state solutions for all sufficiently small $\epsilon> 0$

A second order semilinear neutral functional differential inclusion with nonlocal conditions and multivalued impulse characteristics in a separable Banach space is considered. By developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the topological structure of $C^1$-solution sets is established; and some interesting discussion is offered when the multivalued nonlinearity of the inclusion is a weakly upper semicontinuous map satisfying a condition expressed in terms of the Hausdorff measure.

We consider a singular nonlocal viscoelastic problem with a nonlinear source term and a possible damping term. We prove that if the initial data enter into the stable set, the solution exists globally and decays to zero with a more general rate, and if the initial data enter into the unstable set, the solution with nonpositive initial energy as well as positive initial energy blows up in finite time. These are achieved by using the potential well theory, the modified convexity method and the perturbed energy method.

Using averaging formulas, we compute the Lefschetz, Nielsen and Reidemeister numbers of maps on infra-solvmanifolds modeled on Sol, and we study the Jiang-type property for those infra-solvmanifolds.

Three topics in dynamical systems are discussed. First we deal with cascades and solve two open problems concerning, respectively, product recurrence, and uniformly rigid actions. Next we provide a new example that displays some unexpected properties of strictly ergodic actions of non-amenable groups.

In [Nodal solutions for a class of degenerate boundary value problems, Adv. Nonl. Studies 15 (2015), 253-288], a family of degenerate one dimensional boundary value problems was studied and the existence of positive (and negative) solutions and solutions that possess one interior node was shown for a range of values of a parameter, $\lambda$. It was conjectured that there is a natural extension of these results giving solutions with any prescribed number of interior nodes. This conjecture will be established here.

Let $R$ be a real closed field and let some o-minimal structure extending $R$ be given. Let $F\colon X \rightrightarrows R^m$ be a definable multivalued lower semicontinuous mapping with nonempty definably connected values defined on a definable subset $X$ of $R^n$ of dimension $1$ ($X$ can be identified with a finite graph immersed in $R^n$). Then $F$ admits a definable continuous selection.