Banca de DEFESA: George Demetrios Fernandes Leitão Kiametis

Uma banca de DEFESA de DOUTORADO foi cadastrada pelo programa.
STUDENT : George Demetrios Fernandes Leitão Kiametis
DATE: 23/01/2024
TIME: 14:00
LOCAL: PPGMAT
TITLE:

Some Caffarelli-Kohn-Nirenberg’s type problems in RN

KEY WORDS:

Caffarelli-Kohn-Nirenberg type problems; Berestycki Lions type problem; Existence and concentration of ground state solutions; Existence of positive and nodal ground state solutions.

PAGES: 68
BIG AREA: Ciências Exatas e da Terra
AREA: Matemática
SUMMARY:

IIn this work we prove some results concerning to Caffarelli-Kohn-Nirenberg's type problems in
$\mathbb{R}^N$.
In the first chapter we prove the existence of nontrivial solutions with Berestycki-Lions type nonlinearities. More
precisely, we study the following classes of problems
$$
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)+ |x|^{-bp^{*}}|u|^{p-2}u= |x|^{-bp^{*}} h(u) \ \
\mbox{in} \ \mathbb{R}^{N},
\leqno{(PM)}
$$
and
$$
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)= |x|^{-bp^{*}}f(u) \ \
\mbox{in} \ \mathbb{R}^{N},
\leqno{(ZM)}
$$
where $1<p<N$, $0\leq a< \frac{N-p}{p}$, $a<b\leq a+1$, $p^{*}=p^{*}(a,b)=\frac{pN}{N-dp}$ and $d=1+a-b$.
In the second chapter we prove the existence and concentration of ground state solutions for a class of subcritical,
critical or supercritical Caffarelli-Kohn-Nirenberg type problems. More precisely, we are going to study the following
class of quasilinear problems
\begin{equation*}\tag{$P_{\mu,\varrho,\sigma}$}
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)+|x|^{-bp^{*}}[1+\mu V(z)]|u|^{p-2}u = |x|^{-
bp^{*}}[f(u)+\varrho|u|^{\sigma-2}u]
\end{equation*}
in $\mathbb{R}^{N}$, where $1<p<N$, $0\leq a< \frac{N-p}{p}$, $a<b\leq a+1$, $p^{*}=p^{*}(a,b)=\frac{pN}{N-dp}$,
$d=1+a-b$ and $\mu>0$.
In the third chapter we prove the existence of a positive and a nodal ground state solutions to the following class of
Caffarelli-Kohn-Nirenberg type problems
\begin{equation*}
-\mbox{div}\left(|x|^{-ap}|\nabla u|^{p-2}\nabla u\right)+|x|^{-bp^{*}} V(x)|u|^{p-2}u = |x|^{-
bp^{*}}K(x)f(u) \quad\hbox{in $\mathbb{R}^{N}$}, \quad\hbox{(P)}\
\end{equation*}

\noindent where
$1<p<N$, $0\leq a< \frac{N-p}{p}$, $a<b\leq a+1$, $p^{*}=p^{*}(a,b)=\frac{pN}{N-dp}$ and $d=1+a-b$.
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:
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