This paper is devoted to the study of $L_p$ Lyapunov-typeinequalities for linear systems of equations with Neumann boundaryconditions and for any constant $p \geq 1$. We consider ordinaryand elliptic problems. The results obtained in the linear case arecombined with Schauder fixed point theorem to provide new resultsabout the existence and uniqueness of solutions for resonantnonlinear problems. The proof uses in a fundamental way thenontrivial relation between the best Lyapunov constants and theminimum value of some especial minimization problems.

Let $f\colon M\to M$ be a self-map on a $3$-dimensional flat Riemannian $M$. We compute the Lefschetz number and the Nielsen number of $f$ by using the infra-nilmanifold structure of $M$ and the averaging formulasfor the Lefschetz numbers and the Nielsen numbers of maps on infra-nilmanifolds. For each positive integer $n$, we provide an explicit algorithm for a complete computation of the Nielsen type numbers $N\mathrm{P}_n(f)$ and $N\Phi_n(f)$ of $f^n$.

By using the method of coincidence degree theory and Lyapunovfunctions, some new criteria are established for the existence andglobal exponential stability of anti-periodic solutions to impulsiveCohen-Grossberg neural networks on time scales. Our results are neweven if the time scale $\mathbb{T}=\mathbb{R}$ or $\mathbb{Z}$.Finally, an example is given to illustrate our results.

We prove the existence of multiple nonnegative nonsemitrivialsolutions for a degenerate quasilinear elliptic system. Ourtechnical approach is based on variational methods.

We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle and the Mountain Pass Theorem.

In this paper, a discrete Hematopoiesis model is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique positive periodic solution. Particularly, we give iterative sequence which converges to the positive periodic solution. In addition, the global attractivity of positive periodic solution is also investigated.

In this work we study several questions concerning to abstract fractional Cauchy problems of order $\alpha\in(0,1)$. Concretely, we analyze the existence of local mild solutions for the problem, and its possible continuation to a maximal interval of existence. The case of critical nonlinearities and corresponding regular mild solutions is also studied. Finally, by establishing some general comparison results, we apply them to conclude the global well-posedness of a fractional partial differential equation coming from heat conduction theory.

We consider aseminonlinear Neumann problem driven by the $p$-Laplacian plus an indefinite and unbounded potential. The reaction of the problem is resonant at $\pm \infty$ with respect to the higher parts of the spectrum. Using critical point theory, truncation and perturbation techniques, Morse theory and the reduction method, we prove two multiplicity theorems. One produces three nontrivial smooth solutions and the second four nontrivial smooth solutions.

In this work we obtain an existence result for a class of singular quasilinearelliptic Dirichlet problems on a smooth bounded domain containing the origin. By using a Caffarelli-Kohn-Nirenbergtype inequality, a critical point result fordifferentiable functionals is exploited, in order to prove theexistence of a precise open interval of positive eigenvalues forwhich the treated problem admits at least one nontrivial weak solution. In the case ofterms with a sublinear growth near the origin, we deduce the existence of solutions for small positive values of the parameter. Moreover, the corresponding solutions have smaller and smaller energies as the parameter goes to zero.

In this paper, we study the existence, stability, permanence, andglobal attractor of coexistence states (i.e. the densities of allthe species are positive in $\Omega$) to the following diffusivetwo-competing-prey and one-predator systems with preys competitionand predator saturation:$$\begin{cases} -\Delta\displaystyle u=u\bigg(a_1-u-b_{12}v-\frac{c_1w}{(1+\alpha_1u)(1+\beta_1w)}\bigg)& {\rm in}\ \Omega,\\\displaystyle -\Deltav=v\bigg(a_2-b_{21}u-v-\frac{c_2w}{(1+\alpha_2v)(1+\beta_2w)}\bigg)&{\rm in}\ \Omega,\\\displaystyle -\Deltaw=w\bigg(\frac{e_1u}{(1+\alpha_1u)(1+\beta_1w)}+\frac{e_2v}{(1+\alpha_2v)(1+\beta_2w)}-d\bigg)&{\rm in}\ \Omega,\\k_1\partial_\nu u+u=k_2\partial_\nuv+v=k_3\partial_\nu w+w=0 & {\rm on}\ \partial\Omega, \end{cases}$$where $k_i\geq 0$ $(i=1,2,3)$ and all the other parameters are positive, $\nu$is the outward unit rector on $\partial\Omega$, $u$ and $v$ aredensities of the competing preys, $w$ is the density of the predator.

In this paper we prove the existence of multiple classical solutions for the fourth-order problem$$\begin{cases}\Delta^2 u = \mu u+ u ^{2_* -1} & \text{in } \Omega,\\u,\quad -\Delta u\gt 0 & \text{in } \Omega,\\u,\quad \Delta u = 0 & \text{on } \partial\Omega,\end{cases}$$where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq8$, $2_*=2N/(N-4)$ and $\mu_1(\Omega)$ is the first eigenvalue of $\Delta^2$ in $H^2(\Omega)\cap H_{0}^{1}(\Omega)$. We prove that there exists $0 < \overline{\mu}< \mu_1(\Omega)$ such that, for each $0 < \mu < \overline{\mu}$, the problem has at least ${\rm cat}_{\Omega}(\Omega)$ solutions.

We consider a class of nonlinear, coercive elliptic equations driven by a nonhomogeneous differential operator. Using variational methods together with truncation and comparison techniques, we show that the problem has at least three nontrivial solutions, all with sign information. In the special case of $(p,2)$-equations, using tools from Morse theory, we show the existence of four nontrivial solutions, all with sign information. Finally, for a special class of parametric equations, we obtain multiplicity theorems that substantially extend earlier results on the subject.

We study the existence of ground state andbound state for the following Schrödinger-Poisson equation\begin{equation}\nonumber\begin{cases} -\Delta u + V(x) u+ \lambda\phi (x) u =\mu u+|u|^{p-1}u, & x\in \mathbb{R}^3, \\ -\Delta\phi = u^2, \quad \lim\limits_{|x|\to +\infty}\phi (x)=0,\end{cases}\leqno{(\mathrm{P})}\end{equation}where $p\in(3,5)$, $\lambda > 0$, $V\inC(\mathbb{R}^3,\mathbb{R}^+)$ and $\lim\limits_{|x|\to+\infty}V(x)=\infty$. By using variational method, we prove thatfor any $\lambda > 0$, there exists $\delta_1(\lambda) > 0$ such thatfor $\mu_1 < \mu < \mu_1 + \delta_1(\lambda)$, problem (P) has a nonnegativeground state with negative energy, which bifurcates from zero solution; problem (P) has a nonnegative bound state withpositive energy, which can not bifurcate from zero solution. Here $\mu_1$ is the first eigenvalue of $-\Delta+V$. Infinitely many nontrivial bound states are also obtained withthe help of a generalized version of symmetric mountain passtheorem.

In this paper, using variational methods, we establish the existence and multiplicity of weak solutionsfor nonhomogeneous quasilinear elliptic equations of the form$$-\Delta_n u + a(x)|u|^{n-2}u= b(x)|u|^{n-2}u+g(x)f(u)+\varepsilonh \quad \mbox{in }\mathbb{R}^n ,$$%where $n \geq 2$, $ \Delta_n u \equiv {\rm div}(|\nabla u|^{n-2}\nablau)$ is the $n$-Laplacian and $\varepsilon$ is a positiveparameter. Here the function $g(x)$ may be unbounded in $x$ andthe nonlinearity $f(s)$ has critical growth in the sense ofTrudinger-Moser inequality, more precisely $f(s)$ behaves like$e^{\alpha_0 |s|^{n/(n-1)}}$ when $s\to+\infty$ for some$\alpha_0 > 0$. Under some suitable assumptions and based on aTrudinger-Moser type inequality, our results are proved by usingEkeland variational principle, minimization and mountain-passtheorem.

We prove that, under some reasonable requirements, the unit balls of the spaces $L^p(m)$ and $L^\infty(m)$ of a vector measure of compact range$m$ are compact with respect to the topology $\tau_m$ of pointwiseconvergence of the integrals. This result can be considered as a generalization of the classical Alaoglu Theorem to spaces of $p$-integrablefunctions with respect to vector measures with relatively compactrange. Some applications to the analysis of the Saks spaces defined by the norm topology and $\tau_m$ are given.

We present a computer assisted method for proving the existence of globally attracting fixed points of dissipative PDEs.An application to the viscous Burgers equation with periodic boundary conditions and a forcing function constant in time is presented as a casestudy. We establish the existence of a locally attracting fixed point by using rigorous numerics techniques. To prove that the fixed point is, in fact, globally attracting we introduce a technique relying on a construction of an absorbing set, capturing any sufficiently regular initial condition after a finite time. Then the absorbing set is rigorously integratedforward in time to verify that any sufficiently regular initial condition is in the basin of attraction of the fixed point.

Using a resolvent convergence result from [M.C. Carbinatto and K.P. Rybakowski, Resolvent convergence for Laplace operators on unbounded curved squeezed domains, Topol. Methods Nonlinear Anal. 42 (2013), 233-257] we prove Conley index and index braid continuation results for reaction-diffusion equations on singularly perturbed unbounded curved squeezed domains.