Banca de DEFESA: Romulo Diaz Carlos

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : Romulo Diaz Carlos
DATE: 25/01/2024
TIME: 14:00
LOCAL: Departamento de Matemática
TITLE:

O estudo de problemas não lineares do tipo Kirchhoff-Boussinesq

KEY WORDS:

Biharmonic-p-Laplacian operator, Kirchhoff-Boussinesq problems type, variational methods, critical exponential growth.

PAGES: 80
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

In this thesis, we study the existence and multiplicity of solutions for the following class of
problems
$$
\left\{
\begin{array}{l}
\Delta^{2} u \pm\Delta_p u + V(x)u= f(u) + \beta |u|^{2_{**}-2}u\
\mbox{in} \ \ \Omega, \ \ \\ u\in H^{2}\cap H^{1}_{0}(\Omega),
\end{array}
\right.\leqno{(P_{i})}
$$
where $(P_{i})~(i=1,2,3)$ correspond to the three problems we considered in Chapters 1-3, respectively, $\Omega
\subset \mathbb{R}^N$ is a smooth domain, $\beta \in \{0,1\}$, $2< p< 2^{*}= \frac{2N}{N-2}$ if $N\geq 3$, $2_
{**}= \infty$ if $N=3,4$ and $2_{**}= \frac{2N}{N-4}$ if $N\geq 5$.
%, $\beta \in \{0,1\}$, $f$ is just a continuous function satisfying assumptions that will be stated throughout the text
and $V$ is a continuous function.
%\textcolor{blue}{(I believe you may combine the abstracts of these three published or accepted papers. And you
may refer to the following forms.)}

%\textcolor{red}{(
%In this thesis we investigate the existence and multiplicity results for several kinds of Kirchhoff type problems. The
thesis consists of three chapters. \\
%The Chapter 1 is devoted to \\
%In Chapter 2 we study\\
%Eventually, ***** are proved in Chapter 3. )}
\medskip
The Chapter 1 is devoted to existence result of solutions for the problem $(P_{1})$ when $V=0$ and $\beta =0$,
where $\Omega\subset\mathbb{R}^{4}$ is a smooth bounded domain,
$2< p < 4$ and $f$ is a superlinear continuous function with exponential subcritical or critical growth. We apply the
Nehari manifold method to prove the main results.
\medskip
In Chapter 2 we establish an existence and multiplicity of solutions for the problem $(P_{2})$ when $V=0$ and
$\beta \in \{0,1\}$,
where $\Omega\subset\mathbb{R}^{N}$ is a bounded and smooth domain and $f$ is a continuous function. In this
chapter, we show the existence and multiplicity of nontrivial solutions by using minimization technique on the Nehari
manifold, the Mountain Pass Theorem and Genus theory. The subcritical case $\beta=0$ and the critical case
$\beta=1$ are considered.
\medskip
In Chapter 3 is concerned with the existence of a ground state solution for the problem $(P_{3})$. Here $V$ and $f$
are continuous functions with $V$ being either periodic or asymptotic at infinity to a periodic function. The function
$f$ has subcritical growth and behaves like
$|u|^{q-2}u$ with $p<q< 2_{**}$. Using variational methods, we prove the existence of a ground state solution in the
subcritical case, i.e, $\beta=0$ and the critical case, i.e, $\beta=1$.

COMMITTEE MEMBERS:
Presidente - 1177944 - GIOVANY DE JESUS MALCHER FIGUEIREDO
Interno - 1860632 - LUIS HENRIQUE DE MIRANDA
Interno - 3158033 - MA TO FU
Externo à Instituição - GUSTAVO SILVESTRE DO AMARAL COSTA - UFMA
Externo à Instituição - UBERLANDIO BATISTA SEVERO - UFPB