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Previous issue date: 2018-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In the first two chapters, we consider the following problem
\begin{equation*}
\left \{
\begin{array}{rcll}
\alpha \Delta^{2} u + \beta \Delta u & = & f(x,u)\, & \mbox{in}\,\, \Omega \\
u = \Delta u & = & 0 \, &\mbox{on } \,\,\, \partial \Omega,
\end{array}
\right.
\end{equation*}
where $\displaystyle{\Delta^{2} u = \Delta(\Delta u)-\,\mbox{biharmonic (fourth-order
operator)}}$,
$\alpha > 0$ and $ \beta \in \R.$ The subset $\displaystyle{ \Omega \subset \mathbb{R}^{N}\,
(N \geq 4)}$ is as somooth bounded domain and $\displaystyle{ f \in C(\overline{\Omega}
\times \mathbb{R},\mathbb{R}) }.$ In each of the results obtained, we will consider different
technical hypotheses and characteristics for the nonlinear function $f$ e for the value of the
constant $ \beta. $
In the third chapter, we study an equation of the concave type super linear, of the form:
\begin{equation}
\left \{
\begin{array}{rcll}
\alpha \Delta^{2} u + \beta \Delta u & = & a(x)|u|^{s-2}u + f(x,u)\, & \mbox{in}\,\, \Omega \\
u = \Delta u & = & 0 \, &\mbox{on} \,\,\, \partial \Omega,
\end{array}
\right.
\end{equation}
where $\beta \in (-\infty, \alpha \lambda_{1}).$ We consider that the function $a \in L^{\infty}
(\Omega)$ and $s \in (1,2).$
Finally, in the last chapter we will consider a fourth order problem in which nonlinearity is also of
the convex concave type. More precisely, we study the following class of equations:
\begin{equation}
\left\{ \begin{aligned}
\alpha \Delta^{2} u + \beta \Delta u & = \mu a(x)|u|^{q-2}u + b(x)|u|^{p-2}u&\,\,\,\,\
&\mbox{in}\,\, \Omega \\
u = \Delta u & = 0 & \,\,\,\,&\mbox{on} \,\, \partial \Omega,
\end{aligned}
\right.
\end{equation}
where the parameter $ \mu > 0 $, the powers $ 1 <q <2 <p <2 N / (N - 4) $. In addition we assume
that the functions $ \displaystyle {a, b: \Omega \rightarrow \mathbb {R}}$ are continuous that can
change signal and, $ a ^{+}, b ^{+} \neq 0. $ / Nos dois primeiros Capítulos, consideramos a seguinte classe de problemas:
\begin{equation*}
\left \{
\begin{array}{rcll}
\alpha \Delta^{2} u + \beta \Delta u & = & f(x,u)\, & \mbox{em}\,\, \Omega \\
u = \Delta u & = & 0 \, &\mbox{sobre } \,\,\, \partial \Omega,
\end{array}
\right.
\end{equation*}
onde $\displaystyle{\Delta^{2} u = \Delta(\Delta u)-\,\mbox{biharmônico},}$
$\alpha > 0$ e $ \beta \in \R.$ O subconjunto $\displaystyle{ \Omega \subset
\mathbb{R}^{N}\,(N \geq 4)}$ será um domínio limitado e a não linearidade $\displaystyle{
f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R}) }.$ Em cada um dos resultados
obtidos, consideraremos hipóteses técnicas e características diferentes para a função não
linear $f$ e para o valor da constante $\beta.$
No terceiro Capítulo, estudamos uma equação do tipo côncavo super linear, da forma:
\begin{equation*}
\left \{
\begin{array}{rcll}
\alpha \Delta^{2} u + \beta \Delta u & = & a(x)|u|^{s-2}u + f(x,u)\, & \mbox{em}\,\,
\Omega \\
u = \Delta u & = & 0 \, &\mbox{sobre } \,\,\, \partial \Omega,
\end{array}
\right.
\end{equation*}
onde $\alpha > 0$ e $\beta \in (-\infty, \alpha \lambda_{1})$. Consideramos que a função
$a \in L^{\infty}(\Omega)$ e que $s \in (1,2).$
Por fim, no último Capítulo vamos considerar um problema de quarta ordem no qual a não
linearidade é do tipo côncavo-convexa. Mais precisamente, estudamos a seguinte classe de
equações:
\begin{equation*}
\left\{ \begin{aligned}
\alpha \Delta^{2} u + \beta \Delta u & = \mu a(x)|u|^{q-2}u + b(x)|u|^{p-2}u&\,\,\,\,\
&\mbox{em}\,\, \Omega \\
u = \Delta u & = 0 & \,\,\,\,&\mbox{sobre} \,\, \partial \Omega,
\end{aligned}
\right.
\end{equation*}
onde o parâmetro $\mu > 0$ e as potências $ 1 < q < 2 < p < 2 N /(N - 4)$. Adicionalmente
supomos que as funções $\displaystyle{a, b : \Omega \rightarrow \mathbb{R} }$ sejam
contínuas podendo trocar de sinal em $\Omega$ e que $a^{+},b^{+} \neq 0.$
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.bc.ufg.br:tede/8257 |
| Date | 27 February 2018 |
| Creators | Cavalcante, Thiago Rodrigues |
| Contributors | Silva, Edcarlos Domingos da, Silva, Edcarlos Domingos da, Gonçalves, José Valdo, Carvalho, Marcos Leandro, Santos, Carlos Alberto Pereira, Figueiredo, Giovany de Jesus Malcher |
| Publisher | Universidade Federal de Goiás, Programa de Pós-graduação em Matemática (IME), UFG, Brasil, Instituto de Matemática e Estatística - IME (RG) |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/doctoralThesis |
| Format | application/pdf |
| Source | reponame:Biblioteca Digital de Teses e Dissertações da UFG, instname:Universidade Federal de Goiás, instacron:UFG |
| Rights | http://creativecommons.org/licenses/by-nc-nd/4.0/, info:eu-repo/semantics/openAccess |
| Relation | 6600717948137941247, 600, 600, 600, 600, -4268777512335152015, -713664642194004237, 2075167498588264571 |
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