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In this work, we will use the Mountain Pass Theorem for an even Functional, Genus Theory, Ekeland's variational principle and some properties involving Nehari manifolds to obtain existence and multiplicity of solutions for the following class of quasilinear problems involving variable exponents 8<: p(x)u + jujp(x)2u = f(x; u); x 2 u 2 W1;p(x) 0 ( ) n f0g where is a bounded domain in RN, not necessarily bounded, p(x) is the p(x)-Laplacian operator given by p(x)u = divjrujp(x)2ru; p: ! R and f : R ! R are continuous functions satisfying certain conditions, which will specified be later on.

The notions of connection and covariant derivative has its origin in the field of Riemannian geometry , where there is no distinction between them. In fact, in this study we found that these notions are equivalent if we consider modules over K-algebras of finite type. We also show that the existence of connections implies the existence of covariant derivative. The main goal of this study is to determine which modules admit connections. We easily verified that the projective modules admit connections. In fact, they form an affine space. But we also display a module that is not projective and has connection. Later, inspired by Swan's theorem, we explore in a straightforward way modules formed by sections of the tangent bundle of some surfaces in 3-dimensional real space. Finally, we study the notion of connection introduced by Alain Connes in modules over K-algebras not necessarily commutative. And we find in that context that the modules that have connection are exactly the projectives modules.

In this paper we study the concept of lineability and its recent applications to some sets of continuous real functions. These sets are formed by functions that achieve the absolute maximum in a single point of its domain. In the first chapter we consider the real line and its closed and semi-closed domais as intervals for these functions. In the second chapter we study more general results than those in the previous chapters. In the third chapter we present the theory of degree of continuous applications of Sn in Sn as a tool to demonstrate the Borsuk-Ulam theorem. This result is used a crucial tool in Chapter 2.

In this work, we study qualitative properties of solutions of the semilinear elliptic equation class 8<: u + f(u) = 0, em , u = 0, em @ , defined in different kinds of unbounded domains of Rn, among them, infinite cylinders, half spaces and Lipschitz domains. We analyze properties like convergence, monotonocity and symmetry of solutions of the problem (1), when f satisfy certain conditions suitable. For this purpose, we will use various kinds of maximum principles, the moving planes method,elliptic estimates and compacity theorems. We also studied some results about Schrödinger operators and we prove the De Giorge conjecture in dimension n = 2.

In this work we study the first order partial differential equations on the neighborhood of an isolated zero. Using the classification of singular points put by Izumiya in [27] and [28], we study the multiplicity of such equations introduced in [15]. When the first order partial differential equation defines an implicit differential equation, the definition of multiplicity coincides with the notion of multiplicity introduced by Bruce and Tari in [21]. We will also study the invariance of this multiplicity by smooth equivalence.

This work addresses a class of Trudinger-Moser type inequalities in weighted Sobolev spaces in R2. As an application of these inequalities and by using variational methods, we establish sufficient conditions for the existence, multiplicity and nonexistence of solutions for some classes of nonlinear Schrödinger elliptic equations (and systems of equations) with unbounded, singular or decaying radial potentials and involving nonlinearities with exponential critical growth of Trudinger-Moser type.

In the present work, we study controllability results for two problems on the theory of the partial differential equations. We use global Carleman inequalities to show the null controllability for the heat equation and for a PDE with complex principal part. We obtain the control of minimal norm solving a dual minimization problem.

In this work, we study issues related the existence, nonexistence and regularity of solutions to semilinear Schrödinger equations of type u + a(x)u = jujp2u; u 2 H1(RN); where N 2, p > 2 if N = 2 and 2 < p < 2N=(N 2) if N 3 and the potential a(x) is a positive function that belongs to L1(RN). To obtain the results, we use a Linking Theorem and the Principle of Symmetric Criticality.

In this work, we study the Navier-Stokes flow in R2 8> >>>>>><> >>>>>>: @u @t − ⌫!u + (u ·r)u + rp = f em ⌦ ⇥ [0,+1) , divu = r· u = 0 em ⌦⇥ [0,+1) , u = 0 sobre @⌦ ⇥ [0,+1) , u(·, 0) = u0 em ⌦, in an unbounded domain such that the Poincar´e s inequality is holds, i.e., there is a constant #1 > 0 such that we have the following inequality Z⌦ $2dx  1 #1 Z⌦ |r$|2dx, for all $ 2 H1 0 (⌦). We show the existence of global attractor in the natural phases spaces for this system exploring the energy equation of the problem

In this thesis, we study controllability results of some phenomena modeled by Partial Differential Equations (PDEs): Multi objective control problem, for parabolic equations, following the Stackelber-Nash strategy is considered: for each leader control which impose the null controllability for the state variable, we find a Nash equilibrium associated to some costs. The leader control is chosen to be the one of minimal cost. Null controllability for the linear Schrödinger equation: with a convenient space-time discretization, we numerically construct boundary controls which lead the solution of the Schrödinger equation to zero; using some arguments of Fursikov-Imanuvilov (see [Lecture Notes Series, Vol 34, 1996]) we construct controls with exponential decay at final time. Null controllability for a Schrödinger-KdV system: in this work, we combine global Carleman estimates with energy estimates to obtain an observability inequality. The controllability result holds by the Hilbert Uniqueness Method (HUM). Controllability results for a Euler type system, incompressible, inviscid, under the influence of a temperature are obtained: we mainly use the extension and return methods

In this work, we will establish a version of the Mountain Pass Theorem due to Martin Schechter [12], which will provide a Cerami sequence at a max-min level. As a consequence of this result, together with the Ekeland variational principle, we obtain some results of existence and multiplicity of solution for a class of semilinear elliptic problems involving a nonlinearity of concave-convex type

In this thesis we establish existence and multiplicity results for solutions to some classes of problems on RN involving the p(x)-Laplacian operator. In the first part, we consider classes of problems dealing with nonlinearities possessing critical growth. Ultimately, we consider a class of problems with a nonlinearity possessing a subcritical growth. In this latter case, we searched for multi-bump solutions. Among the tools we used are Mountain Pass Theorem, Concentration-Compactness Principle, Lion s Lemma, Ekeland s Variational Principle and Penalization Method

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES

In this work we investigate monotonicity and symmetry properties of of solutions to equations involving the p-Laplace-Beltrami operator in hyperbolic space and sphere. The main tools used to obtain the result is a variant of the method of moving planes and a careful use of the maximum and comparison principles

In this work we prove results of existence, multiplicity and blow-up solutions for a boundary value problem originated Corrosion Modeling and involving a parameter > 0. We obtain the existence of an infinity of solutions of the problem using the so-called theory of Lyusternik-Schnirelman, and ideas due to S.I. Pohozaev and A. Bahri. The basis of our analysis of limite behavior, Blow-up, is a uniform estimate @v @n in L1(@ ) where v is the solution for the parameter , combined with an adaptation of techniques developed by Brezis and Merle. Precisely, we prove that when ! 0+ our solutions, from to a subsequence, develops a finite number of singularities on @

The main goal of this work is to furnish a proof of the well-known Rentschler s Theorem, which describes the structure of the locally nilpotent derivations on the polynomial ring in two indeterminates (over a field of characteristic zero), up to conjugation by tame automorphisms. As a central application of this result, we prove Jung s Theorem, concerning the generators of the group of automorphisms in two variables. Finally, some examples are discussed, illustrating connections to other important topics.

In this work, we will use the Mountain Pass Theorem, Ekeland s Variational Principle, the Concentration-Compactness Principle, the Brezis & Nirenberg Method, Penalization Method and some properties involving Nehari manifolds to obtain existence and multiplicity of solutions for the following class of elliptic systems. () 8<: u + V (x)u + u = r(x; u) em R3; = u2 em R3; where r : R3 R ! R is a function that has critical growth.

In this work, we study the existence of positive solutions for a class of semilinear elliptic equations in a smooth bounded domain, with Dirichlet boundary condition and non-linear terms changing sign as well as with small perturbations. In order to obtain the positive solution, in the first case we use a version of the Mountain Pass Theorem in Ordered Banach spaces. In the second case, the main term is under assumptions that guarantee the application of the usual Mountain Pass Theorem and the perturbation term does not require any hypothesis.

Quantum Calculus consists of a different approach to the subject Calculus, which is commonly studied in courses such as Mathematics and Physics. We therefore chose to address this theme, aiming to present the q-derivative and its applications, as well as quantum integration and its applications. For this, we rely on principles such as q-calculus and h-calculus, which consist of two topics of Quantum Calculus.

The Bohnenblust{Hille inequality guarantees the existence of a function C : N ! [1;+1), corresponding to each positive integer m, a constant C(m) with the following property: regardless of the choice of the natural N and the bounded m-linear form U : `N 1 `N 1 ! K, the sequence (U(ei1 ; : : : ; eim))N i1;:::;im=1 belongs to ` 2m m+1 and its 2m m+1-norm is bounded by C(m)kUk, where k k denotes the supremum norm. Apart from the intrinsic mathematical interest, for C(m) does not depend on each natural N, the diversity and relevance of the applications enrich the result further. On the actual scenario, recent explicit estimates for the constants C(m) present optimal asymptotic behaviour and subexponencial growth, in contrast to the exponential growth of the known estimates from the last decades. Valuable informations concerning the optimal constants (the lowest ones with the previous property stated) emerge, once proved that these also enjoy of an optimal asymptotic growth, if it exists.