In this thesis, direct and inverse problems concerning nodal solutions associated with the one-dimensional p-Laplacian operators are studied. We first consider the eigenvalue problem on (0, 1), −(y0(p−1))0 + (p − 1)q(x)y(p−1) = (p − 1) £fw(x)y(p−1) (0.1) Here f(p−1) := |f|p−2f = |f|p−1 sgn f. This problem, though nonlinear and degenerate, behaves very similar to the classical Sturm-Liouville problem, which is the special case p = 2. The spectrum {£fk} of the problem coupled with linear separated boundary conditions are discrete and the eigenfunction yn corresponding to£fn has exactly n−1 zeros in (0, 1). Using a Pr¡Lufer-type substitution and properties of the generalized sine function, Sp(x), we solve the reconstruction and stablity issues of the inverse nodal problems for Dirichlet boundary conditions, as well as periodic/antiperiodic boundary conditions whenever w(x) £f 1. Corresponding Ambarzumyan problems are also solved. We also study an associated boundary value problem with a nonlinear nonhomogeneous term (p−1)w(x) f(y(x)) on the right hand side of (0.1), where w is continuously differentiable and positive, q is continuously differentiable and f is positive and Lipschitz continuous on R+, and odd on R such that f0 := lim y!0+ f(y) yp−1 , f1 := lim y!1 f(y) yp−1 . are not equal. We extend Kong¡¦s results for p = 2 to general p > 1, which states that whenever an eigenvalue _n 2 (f0, f1) or (f1, f0), there exists a nodal solution un having exactly n − 1 zeros in (0, 1), for the above nonhomogeneous equation equipped with any linear separated boundary conditions. Although it is known that there are indeed some differences, Our results show that the one-dimensional p-Laplacian operator is still very similar to the Sturm-Liouville operator, in aspects involving Pr¡Lufer substitution techniques.

In this dissertation, we study the existence and nonexistence of positive radial solutions for classes of quasilinear elliptic equations and systems in a ball with Dirichlet boundary conditions. Our nonlinearities are asymptotically p-linear at infinity and are allowed to be singular at zero with non-positone structure, which have not been considered in the literature. In the one parameter single equation problem, we are able to show the existence of a positive radial solution with precise lower bound estimate for a certain range of the parameter. We also extend the study to a class of asymptotically p-linear system with two parameters and in the presence of singularities. We establish the existence of a positive solution with a precise lower bound estimate when the product of the parameters is in a certain range. Necessary and sufficient conditions for the existence of a positive solution are also obtained for both the single equation and system under additional assumptions. Our approach is based on the Schauder Fixed Point Theorem.

In this report the connection between p-harmonic functions and the tug-of-war game is presented and some applications are mentioned. Moreover, sufficient background information of solutions to the p-Laplace equation is given. And to finish, an example of the game is given with simulations. / I denna rapport presenteras kopplingen mellan p-harmoniska funktioner och spelet tug-of-war och några applikationer nämns. Utöver så ges tillräckligt med infomration om lösningar till p-Laplace ekvation. Som avslutning, ges ett exempel på spelet tillsammans med simuleringar.

In this work we study the Generalized Lane-Emden equation and the interplay between the exponents involved and their consequences on the existence and non existence of radial solutions on a unit ball in n dimensions. We extend the analysis to the phase plane for a clear understanding of the behavior of solutions and the relationship between their existence and the growth of nonlinear terms, where we investigate the critical exponent p and a sub-critical exponent, which we refer to as ^p. We discover a structural change of solutions due the existence of this sub-critical exponent which we relate to the same change in behavior of the Lane- Emden equation solutions, for ; = 0; andp = 2, due to the same sub-critical exponent. We hypothesize that this sub-critical exponent may be related to a weighted trace embedding.

Orientador: Simone Mazzini Bruschi / Banca: Cláudia Buttarello Gentile / Banca: Marta Cilene Gadoti / Resumo: Neste trabalho caracterizamos o conjunto dos pontos de equilíbrio de um problema parabólico quasilinear governado pelo p-Laplaciano, p > 2, e do problema parabólico governado pelo Laplaciano / Abstract: In this work we give a characterization set of the equilibrium points of a parabolic problem quasi-linear governed by the p-Laplacian, p > 2, and the a parabolic problem governed by the Laplacian / Mestre

The function $sin x$ as one of the six trigonometric functions is fundamental in nearly every branch of mathematics, and its applications. In this thesis, we study an integral equation related to that of $sin x$: $mbox{~for~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}] mbox{~and~} p>1$ $$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt.$$ Here $hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dt.$ We find that the function $S_{p}(x)$ is well defined. Its properties are also similar to those of $sin x$ : differentiation, identities, periodicity, asymptotic expansions, $cdots$, etc. For example, we have $$|S_{p}(x)|^{p}+|S'_{p}(x)|^{p}=1mbox{~~and~~}frac{d}{dx}(|S'_{p}(x)|^{p-2}S'_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x).$$ We call $S_{p}(x)$ the generalized sine function. Similarly, we define the generalized cosine function $C_{p}(x)$ by $|x|=int_{C_{p}(x)}^{1}(1- t^{p})^{-frac{1}{p}}dt$ for $xin[-frac{hat{pi}_{p}}{2}$,~$frac{hat{pi}_{p}}{2}]$ and derive its properties. Thus we obtain two sets of trigonometric functions: egin{itemize} item[(i)]$~S_{p}(x),~ S'_{p}(x),~ T_{p}(x)=frac{S_{p}(x)}{S'_{p}(x)},~RT_{p}(x)=frac{S'_{p}(x)}{S_{p}(x)},~ SE_{p}(x)=frac{1}{S'_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$ item[(ii)]$~C_{p}(x),~ C'_{p}(x),~RCT_{p}(x)=-frac{C'_{p}(x)}{C_{p}(x)},~ CT_{p}(x)=-frac{C_{p}(x)}{C'_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~ CS_{p}(x)=-frac{1}{C'_{p}(x)}mbox{~¡C~}$ end{itemize}These two sets of functions have similar differentiation formulas, identities and periodic properties as the classical trigonometric functions. They coincide when $p=2$. Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions.

Abstract not available. / text

Made available in DSpace on 2014-06-11T19:24:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-05-07Bitstream added on 2014-06-13T20:07:58Z : No. of bitstreams: 1 martins_j_me_rcla.pdf: 650990 bytes, checksum: 197553843285b3dcdd899cddf89d0ab2 (MD5) / Universidade Estadual Paulista (UNESP) / Neste trabalho caracterizamos o conjunto dos pontos de equilíbrio de um problema parabólico quasilinear governado pelo p-Laplaciano, p > 2, e do problema parabólico governado pelo Laplaciano / In this work we give a characterization set of the equilibrium points of a parabolic problem quasi-linear governed by the p-Laplacian, p > 2, and the a parabolic problem governed by the Laplacian

We study positive solutions to nonlinear elliptic systems of the form: \begin{eqnarray*} -\Delta u =\lambda f(v) \mbox{ in }\Omega\\-\Delta v =\lambda g(u) \mbox{ in }\Omega\\\quad~~ u=0=v \mbox{ on }\partial\Omega \end{eqnarray*} where $\Delta u$ is the Laplacian of $u$, $\lambda$ is a positive parameter and $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial\Omega$. In particular, we will analyze the combined effects of the nonlinearities on the existence and multiplicity of positive solutions. We also study systems with multiparameters and stronger coupling. We extend our results to $p$-$q$-Laplacian systems and to $n\times n$ systems. We mainly use sub- and super-solutions to prove our results.

In this dissertation, we study the existence and multiplicity of positive solutions to classes of one-dimensional singular p-Laplacian problems with nonlinear and intergral boundary conditions when the reaction termis p-superlinear or p-sublinear at infinity. In the p-superlinear case, we prove the existence of a large positive solution when a parameter is small and if, in addition, the reaction term satisfies a concavity-like condition at the origin, the existence of two positive solutions for a certain range of the parameter. In the p-sublinear case, we establish the existence of a large positive solution when a parameter is large. We also investigate the number of positive solutions for the general PHI-Laplacian with nonlinear boundary conditions when the reaction term is positive. Our results can be applied to the challenging infinite semipositone case and complement or extend previous work in the literature.Our approach depends on Amann's fixed point in a Banach space, degree theory, and comparison principles.