We generalize the Bartsch-Li's splitting lemma at infinity for $C^2$-functionals in [T. Bartsch and S.-J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997), no. 3, 419-441] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Different from the previous flow methods our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [D.M. Duc, T.V. Hung and N.T. Khai, Morse-Palais lemma for nonsmooth functionals on normed spaces, Proc. Amer. Math. Soc. 135 (2007), no. 3., 921-927] with some techniques from [M. Jiang, A generalization of Morse lemma and its applications, Nonlinear Anal. 36 (1999), 943-960], [I.V. Skrypnik, Nonlinear Elliptic Equations of a Higher Order, Naukova Dumka, Kiev (1973)], [S.A. Vakhrameev, Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems, J. Sov. Math. 67 (1993), no. 1, 2713-2811]. A simple application is also presented.

In this paper, we deal with the existence of periodic solutions of a kind of Liénard equations with two deviating arguments $$ x''+f(t, x(t-\sigma(t)))x'(t)+g(t, x(t-\tau (t)))=p(t). $$ Some new results on the existence of periodic solutions of the given equations are proved by using the continuation theorem.

The paper provides an existence principle for the Sturm-Liouville boundary value problem with state-dependent impulses \begin{gather*} z''(t) = f(t,z(t),z'(t)) \quad \text{for a.e. } t \in [0,T] \subset \mathbb R, \\ z(0) - az'(0) = c_1, \quad z(T) + bz'(T) = c_2, \\ z(\tau_i+) - z(\tau_i) = J_i(\tau_i,z(\tau_i)), \quad z'(\tau_i+) - z'(\tau_i-) = \mathcal M_i(\tau_i,z(\tau_i)), \end{gather*} where the points $\tau_1, \ldots, \tau_p$ depend on $z$ through the equations \begin{equation*} \tau_i = \gamma(z(\tau_i)), \quad i = 1,\ldots,p, \ p \in \mathbb N. \end{equation*} Provided $a,b \in [0,\infty)$, $c_j \in \mathbb R$, $j = 1,2$, and the data functions $f,J_i,\mathcal M_i$, $i=1,\ldots,p$, are bounded, transversality conditions for barriers $\gamma_i$, $i = 1,\ldots,p$, which yield the solvability of the problem, are delivered. An application to the problem with unbounded data functions is demonstrated.

We discuss S. Park's abstract convex spaces and their relevance to classical convexieties and $L^{\ast}$-operators. We construct an example of a space satisfying the partial KKM principle that is not a KKM space. The existence of such a space solves a problem by S. Park.

In this paper we provide sufficient conditions for the existence of solutions to two-point boundary value problems for nonlinear ordinary differential equations subject to impulses. Our results depend on properties of the nonlinearities as well as on the solution space of the associated linear problem. Our approach is based on topological degree arguments in conjunction with the Lyapunov-Schmidt procedure.

In this paper we establish the existence of solutions for elliptic equations of the form $-{\rm div}(|\nabla u|^{n-2}\nabla u) + V(x)|u|^{n-2}u=g(x,u)+\lambda h$ in $\mathbb{R}^n$ with $n\geq2$. Here the potential $V(x)$ can change sign and the nonlinearity $g(x,u)$ is possibly discontinuous and may exhibit exponential growth. The proof relies on the application of a fixed point result and a version of the Trudinger-Moser inequality.

This paper studies the trajectory behavior of the convective Brinkman-Forchheimer equations in three-dimensional (3D) bounded domains. We first prove the existence of the trajectory attractor ${\mathcal A}^{\rm tr}_\alpha$ for the natural translation semigroup in the trajectory space. Then we establish that the trajectory attractor $\mathcal{A}^{\rm tr}_\alpha$ converges, as $\alpha\rightarrow 0^+$, to the trajectory attractor $\mathcal{A}^{\rm tr}_0$ of the 3D Navier-Stokes equations in a proper topological space.

The main goal of this work is to prove Strauss- and Lions-type results for Orlicz-Sobolev spaces. After, we use these results to study the existence of solutions for a class of quasilinear problems in $\mathbb{R}^{N}$.

In this paper, we consider the Schrödinger equation with a nonlinearity in the critical growth. The purpose of this paper is to establish the existence of ground states via variational methods.

A Connection Matrix Theory approach is presented for Morse-Bott flows $\varphi$ on smooth closed $n$-manifolds by characterizing the set of connection matrices in terms of Morse-Smale perturbations. Further results are obtained on the effect on the set of connection matrices $\mathcal{CM}(S)$ caused by changes in the partial orderings and in the Morse decompositions of an isolated invariant set $S$.

In this work we use the theory of exterior spaces to construct a $\check{C}_{0}^{\mathbf{r}}$-completion and a $\check{C}_{0}^{\mathbf{l}}$-completion of a dynamical system. If $X$ is a flow, we construct canonical maps $X\to \check{C}_{0}^{\mathbf{r}}(X)$ and $X\to \check{C}_{0}^{\mathbf{l}}(X)$ and when these maps are homeomorphisms we have the class of $\check{C}_{0}^{\mathbf{r}}$-complete and $\check{C}_{0}^{\mathbf{l}}$-complete flows, respectively. In this study we find out many relations between the topological properties of the completions and the dynamical properties of a given flow. In the case of a complete flow this gives interesting relations between the topological properties (separability properties, compactness, convergence of nets, etc.) and dynamical properties (periodic points, omega limits, attractors, repulsors, etc.).

The paper deals with equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a solution for them using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, one can derive an existence theorem for the fractional Laplacian, finding solutions of the equation $$ \begin{cases} (-\Delta)^s u=f(x,u) & {\mbox{in }} \Omega,\\ u=0 & {\mbox{in }} \mathbb{R}^n\setminus \Omega, \end{cases} $$ where the nonlinear term $f$ satisfies a linear growth condition.

In this paper, we are concerned with the following quasilinear elliptic equation with concave and convex terms \begin{equation} -\Delta u-{\frac12}\,u\Delta(|u|^2)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in \Omega, \tag($\rm P$) \end{equation} where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $1< p< 2$, $4< q\leq 22^*$. The existence of infinitely many solutions is obtained by the perturbation methods.