TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

 Vol. 45, No. 2           June 2015

Title and Author(s) Page
Matrix Lyapunov inequalities for ordinary and elliptic partial differential equations
 ABSTRACT. This paper is devoted to the study of $L_p$ Lyapunov-type inequalities for linear systems of equations with Neumann boundary conditions and for any constant $p \geq 1$. We consider ordinary and elliptic problems. The results obtained in the linear case are combined with Schauder fixed point theorem to provide new results about the existence and uniqueness of solutions for resonant nonlinear problems. The proof uses in a fundamental way the nontrivial relation between the best Lyapunov constants and the minimum value of some especial minimization problems.
309
The Nielsen type numbers for maps on a 3-dimensional flat Riemannian manifold
Ku Yong Ha and Jong Bum Lee
 ABSTRACT. 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 formulas for 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 $\NP_n(f)$ and $N\Phi_n(f)$ of $f^n$.
327
Global exponential stability and existence of anti-periodic solutions to impulsive Cohen-Grossberg neural networks on time scales
Yongkun Li and Tianwei Zhang
 ABSTRACT. By using the method of coincidence degree theory and Lyapunov functions, some new criteria are established for the existence and global exponential stability of anti-periodic solutions to impulsive Cohen-Grossberg neural networks on time scales. Our results are new even if the time scale $\mathbb{T}=\mathbb{R}$ or $\mathbb{Z}$. Finally, an example is given to illustrate our results.
363
Multiple nonsemitrivial solutions for a class of degenerate quasilinear elliptic systems
 ABSTRACT. We prove the existence of multiple nonnegative nonsemitrivial solutions for a degenerate quasilinear elliptic system. Our technical approach is based on variational methods.
385
Existence of solutions for a class of p(x)-laplacian equations involving a concave-convex nonlinearity with critical growth in R^{N}
Claudianor O. Alves and Marcelo C. Ferreira
 ABSTRACT. 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.
399
Existence and global attractivity of the unique positive periodic solution for discrete hematopoiesis model
Zhijian Yao
 ABSTRACT. 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.
423
Semilinear fractional differential equations: global solutions, critical nonlinearities and comparison results
Bruno de Andrade, Alexandre Nolasco de Carvalho, Paulo M. Carvalho-Neto and Pedro Marín-Rubio
 ABSTRACT. 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.
439
Resonant Neumann equations with indefinite linear part
Giuseppina Barletta, Roberto Livrea and Nikolaos S. Papageorgiou
 ABSTRACT. 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.
469
Bifurcation analysis of a singular elliptic problem modelling the equilibrium of anisotropic continuous media
Giovanni Molica Bisci and Vicentiu D. Radulescu
 ABSTRACT. In this work we obtain an existence result for a class of singular quasilinear elliptic Dirichlet problems on a smooth bounded domain containing the origin. By using a Caffarelli-Kohn-Nirenberg type inequality, a critical point result for differentiable functionals is exploited, in order to prove the existence of a precise open interval of positive eigenvalues for which the treated problem admits at least one nontrivial weak solution. In the case of terms 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.
493
Coexistence states of diffusive predator-prey systems with preys competition and predator saturation
Jun Zhou
 ABSTRACT. In this paper, we study the existence, stability, permanence, and global attractor of coexistence states (i.e. the densities of all the species are positive in $\Omega$) to the following diffusive two-competing-prey and one-predator systems with preys competition and predator saturation: -\Delta u=u\bigg(a_1-u-b_{12}v-\frac{c_1w}{(1+\alpha_1u)(1+\beta_1w)}\bigg) & {\rm in}\ \Omega, -\Delta v=v\bigg(a_2-b_{21}u-v-\frac{c_2w}{(1+\alpha_2v)(1+\beta_2w)}\bigg) &{\rm in}\ \Omega, -\Delta w=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_\nu v+v=k_3\partial_\nu w+w=0 & {\rm on}\ \partial\Omega, 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 $v$ are densities of the competing preys, $w$ is the density of the predator.
509
A fourth-order equation with critical growth: the effect of the domain topology
Jessyca Lange Fereira Melo and Ederson Moreira dos Santos
 ABSTRACT. In this paper we prove the existence of multiple classical solutions for the fourth-order problem \Delta^2 u = \mu u+ u ^{2_* -1} & \text{in } \Omega, u,\quad -\Delta u> 0 & \text{in } \Omega, u,\quad \Delta u = 0 & \text{on } \partial\Omega, where $\Omega$ is a smooth bounded domain in $\rn$, $N\geq8$, $2_*=2N/(N-4)$ 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 $\cat_{\Omega}(\Omega)$ solutions.
551
Nikolaos S. Papageorgiou and Vicentiu D. Radulescu
 ABSTRACT. 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.
575
Standing waves for nonlinear Schrödinger-Poisson equation with high frequency
Jianqing Chen, Zhengping Wang and Xiaoju Zhang
 ABSTRACT. We study the existence of ground state and bound state for the following Schrödinger-Poisson equation -\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, \leqno{(\rom{P})} where $p\in(3,5)$, $\lambda > 0$, $V\in C(\mathbb{R}^3,\mathbb{R}^+)$ and $\lim\limits_{|x|\to +\infty}V(x)=\infty$. By using variational method, we prove that for any $\lambda > 0$, there exists $\delta_1(\lambda) > 0$ such that for $\mu_1 < \mu < \mu_1 + \delta_1(\lambda)$, problem (P) has a nonnegative ground state with negative energy, which bifurcates from zero solution; problem (P) has a nonnegative bound state with positive 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 with the help of a generalized version of symmetric mountain pass theorem.
601
Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in R^n
Manassés de Souza and João Marcos do Ó and Tarciana Silva
 ABSTRACT. In this paper, using variational methods, we establish the existence and multiplicity of weak solutions for 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)+\varepsilon h \quad \mbox{in }\mathbb{R}^n , where $n \geq 2$, $\Delta_n u \equiv \dive(|\nabla u|^{n-2}\nabla u)$ is the $n$-Laplacian and $\varepsilon$ is a positive parameter. Here the function $g(x)$ may be unbounded in $x$ and the nonlinearity $f(s)$ has critical growth in the sense of Trudinger-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 a Trudinger-Moser type inequality, our results are proved by using Ekeland variational principle, minimization and mountain-pass theorem.
615
Compactness in spaces of p-integrable functions with respect to a vector measure
Pilar Rueda and Enrique A. Sánchez-Pérez
 ABSTRACT. 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 pointwise convergence of the integrals. This result can be considered as a generalization of the classical Alaoglu Theorem to spaces of $p$-integrable functions with respect to vector measures with relatively compact range. Some applications to the analysis of the Saks spaces defined by the norm topology and $\tau_m$ are given.
641
Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof
Jacek Cyranka
 ABSTRACT. 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 case study. 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 integrated forward in time to verify that any sufficiently regular initial condition is in the basin of attraction of the fixed point.
655
Curved squeezing of unbounded domains and tail estimates
Krzysztof P. Rybakowski
 ABSTRACT. Using a resolvent convergence result from [7] we prove Conley index and index braid continuation results for reaction-diffusion equations on singularly perturbed unbounded curved squeezed domains.
699