We consider a fairly general class of dynamical systems under assumptions that guarantee the existence of a Lyapunov function around a nontrivial stationary point. Moreover, we prove the existence of a heteroclinic trajectory. Finally, using geometric and topological reasoning, we establish an upper bound for this trajectory. The results can be interpreted as limit theorems in terms of integrated densities for astrophysical models in both relativistic and classical frameworks. These models encompass the static Smoluchowski-Poisson and Tolman-Oppenheimer-Volkoff equations.

We discuss the existence and nonexistence of positive decaying solutions for a semilinear elliptic systems considered in an exterior domain. Applying the subsolutions and supersolutions method and the Sattinger's iteration schema we prove that our problem possesses solutions with minimal growth and finite energy in a neighborhood of infinity. We also formulate necessary conditions for the existence of such solutions for a certain class of nonlinearities. Finally, some nonexistence results are formulated.

In this paper, we study a nonlocal logistic equation with nonlinear advection term \begin{equation}\label{Pp} \begin{cases} \displaystyle -\Delta u+\overrightarrow {\alpha}(x)\cdot {\nabla (u^p)} =\left(\lambda-\int_{\Omega}K(x,y)u^{\gamma}(y)dy \right)u &\mbox{in }\Omega,\\ u=0 &\mbox{ on }\partial\Omega, \end{cases} \tag{\rom{P}$_p$} \end{equation} where $\Omega\subset\R^N$, $N\geq1$, is a bounded domain with smooth boundary, $\overline{\alpha}(x)=(\alpha_1(x),\dots,\alpha_N(x))$ is a flow satisfying suitable condition, $\gamma> 0$, $p\geq1$, $\lambda\in\R$ and $K\colon \Omega\times\Omega\rightarrow\R$ is a nonnegative function with $K\in L^{\infty}(\Omega\times\Omega)$ and verifying other conditions that will be detailed below. It is very important to note that, this equation is not the classic logistic equation due to the inclusion of the term $\overline{\alpha}(x)\cdot \nabla (u^p)$, moreover, the inclusion of the integral nonlocal term on the right-hand side makes the problem closer to a real world situation.

The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set or a $W$-set in a manifold. These are certain classes of compact invariant sets in whose vicinity the asymptotic behaviour of the flow is somewhat controlled. We are mainly concerned with global properties of the dynamics and establish cohomological relations between the non-saddle set and the manifold. As a consequence we obtain a dynamical classification of surfaces (orientable and non-orientable). We also examine robustness and bifurcation properties of non-saddle-sets and study in detail the behavior of $W$-sets in 2-manifolds.

We establish the existence, concentration, and exponential decay of a family of sign-changing solutions for a problem involving exponential critical growth, described by the equation: $$ \begin{cases} -\epsilon^{2}\mbox{div}\left(a(x)\nabla u\right)+V(x)u =K(x) f(u) & \mbox{in $\mathbb{R}^2,$}\\ u \in H^{1}\big(\mathbb{R}^{2}\big). \end{cases} \leqno{(\rom{P}_{\epsilon})} $$ To address the lack of compactness and the competition between the potentials, we employ variational methods alongside a suitable truncation of the nonlinearity.

In this paper we are interested in the study of a two-phase problem equipped with the $\Phi$-Laplacian operator $$ \Delta_\Phi u \coloneqq \mbox{div} \bigg(\varphi(|\nabla u|) \dfrac{\nabla u}{|\nabla u|}\bigg), $$% where $\Phi(s)=e^{s^2}-1$ and $\varphi=\Phi'$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the free boundaries of these minimizers have locally finite perimeter and Hausdorff dimension at most $(N-1)$.

This paper studies directed motion planning algorithms on a CW complex, or more generally, a space decomposed by a poset. We demonstrate that a regular CW complex admits a global directed motion planning algorithm that is compatible with the face poset. Furthermore, as a non-regular case, we compute the directed topological complexity of the directed circle associated with the minimal cell decomposition.

In this paper, we will study the following nonlinear elliptic equation with zero mass on the lattice graph \begin{equation}\label{A}\tag{A} \begin{cases} -\Delta_{p} u= K(x)f(u) & \hbox{in } \mathbb{Z}^N, \\ u\in D^{1, p}\big(\mathbb{Z}^N\big), \end{cases} \end{equation} where $ N\geq 3$, $1< p< N$, $K$ is a nonnegative potential function, $f$ is a continuous function with quasicritical growth or supercritical growth. By employing variational methods, we establish the existence of ground states for the above equation with an asymptotically periodic potential and vanishing potential at infinity. For the case of asymptotically periodic potential, we also generalize the main result from $\mathbb{Z}^N$ to quasi-transitive graphs.

In this paper, we address both a combinatorial continuation question via Lyapunov graphs and the attainability of a preassigned level set, dubbed ground level set and given by its Betti numbers, within a morsification process of a dynamical configuration. The novelty introduced here is a componentwise Lyapunov graph morsification that keeps track of level sets during the morsification process subject to the minimality of the total number of singularities of a morsified flow. The algorithm behind the morsification translates into a system of linear equations whose feasibility is linked to that of a set of inequalities, called componentwise Poincaré-Hopf inequalities, involving the input data. This investigation combines techniques from both homological Conley index and network flow theories.

This article gives a correct reference to the fact that $k_0(H)> 4.58$, where $H$ is an infinite dimensional Hilbert space, to correct a citation in the introduction section of \cite{intrakul:iampiboonvatana}.