Eugénio M. Rocha

On nonlinear parametric problems for p-Laplacian-like operators

We study a nonlinear parametric problem driven by a p-Laplacian-like operator (which need not be homogeneous) and with a ( p - 1)-superlinear nonlinearity which satisfy weaker conditions than the Ambrosetti-Rabinowitz condition. Using critical point theory, we show that for every λ > 0, the nonlinear parametric problem has a nontrivial solution. Then, by strengthening the conditions on the operator and the nonlinearity, and using variational methods together with suitable truncation techniques and tools from Morse theory, we show that, for every λ > 0, the nonlinear parametric problem has three nontrivial smooth solutions.ResumenEn este artículo estudiamos un problema paramétrico no lineal que involucra al operador de tipo p-Laplaciano (que en general no és homogéneo) y donde la derivada del potencial es una función ( p - 1)-superlinear que verifica una condición más débil que la conocida condición de Ambrosetti-Rabinowitz. Utilizando métodos variacionales, mostramos que, para todo λ > 0, el problema paramétrico no lineal tiene una solución no trivial. Entonces, fortaleciendo las condiciones y usando herramientas de la teoría de Morse junto con adecuadas técnicas de truncación, mostramos que, para cada λ > 0, el problema tiene tres soluciones suaves.

A MULTIPLICITY THEOREM FOR A VARIABLE EXPONENT DIRICHLET PROBLEM

We consider a nonlinear Dirichlet problem driven by the p (ċ)-Laplacian. Using variational methods based on the critical point theory, together with suitable truncation techniques and the use of upper-lower solutions and of critical groups, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive, the other negative). The hypotheses on the nonlinearity incorporates in our framework of analysis, both coercive and noncoercive problems.

On computation of the logarithm of the Chen-Fliess series for nonlinear systems

We obtain expressions for the logarithm of the Chen-Fliess series for a nonlinear control system. This logarithm provides an alternative to Sussmann’s exponential product expansion of the Chen-Fliess series. We also formulate a rule for generating the coefficient of any Lie brackets in the logarithm expansion.

A Class of Sub-elliptic Equations on the Heisenberg Group and Related Interpolation Inequalities

We firstly prove the existence of least energy solutions to a class of sub–elliptic equations on the Heisenberg group. Then we use this least energy solution to give a sharp estimate to the smallest positive constant in the Gagliardo–Nirenberg inequality on the Heisenberg group. Finally we point out some extensions to the quasilinear sub–elliptic case.

The effect of immigrant communities coming from higher incidence tuberculosis regions to a host country

We introduce a new tuberculosis (TB) mathematical model, with 25 state-space variables where 15 are evolution disease states (EDSs), which generalises previous models and takes into account the (seasonal) flux of populations between a high incidence TB country (A) and a host country (B) with low TB incidence, where (B) is divided into a community (G) with high percentage of people from (A) plus the rest of the population (C). Contrary to some beliefs, related to the fact that agglomerations of individuals increase proportionally to the disease spread, analysis of the model shows that the existence of semi-closed communities are beneficial for the TB control from a global viewpoint. The model and techniques proposed are applied to a case-study with concrete parameters, which model the situation of Angola (A) and Portugal (B), in order to show its relevance and meaningfulness. Simulations show that variations of the transmission coefficient on the origin country has a big influence on the number of infected (and infectious) individuals on the community and the host country. Moreover, there is an optimal ratio for the distribution of individuals in (C) versus (G), which minimizes the reproduction number

On a class of N-dimensional anisotropic Sobolev inequalities

R_0

Existence of Solutions of a Class of Nonlinear Singular Equations in Lorentz Spaces

R0. Such value does not give the minimal total number of infected individuals in all (B), since such is attained when the community (G) is completely isolated (theoretical scenario). Sensitivity analysis and curve fitting on