CPAA
Existence of positive solutions for $p$--Laplacian problems with weights
Friedemann Brock Leonelo Iturriaga Justino Sánchez Pedro Ubilla
Communications on Pure & Applied Analysis 2006, 5(4): 941-952 doi: 10.3934/cpaa.2006.5.941
We study the existence of positive solutions of the singular quasilinear elliptic equation

-div$(|x|^{-a p}|\nabla u|^{p-2}\nabla u)=|x|^{-(a+1)p+c}f(x,u)$ in $\Omega$

$u=0 $ on $\partial\Omega,$

where $p>1$. We use upper and lower--solutions methods, variational techniques and regularity theory.

keywords: Singular elliptic equations Caffarelli-Kohn-Nirenberg inequality. regularity lower and upper solutions
CPAA
Positive radial solutions for some quasilinear elliptic systems in exterior domains
João Marcos do Ó Sebastián Lorca Justino Sánchez Pedro Ubilla
Communications on Pure & Applied Analysis 2006, 5(3): 571-581 doi: 10.3934/cpaa.2006.5.571
We use fixed-point theorem of cone expansion/compression type to prove the existence of positive radial solutions for the following class of quasilinear elliptic systems in exterior domains

$-\Delta_p u = k_1(|x| )f(u,v),$ for $|x| > 1$ and $x \in \mathbb R^N, $

$-\Delta_p v = k_2(|x|)g(u,v),$ for $|x| > 1 $ and $x \in \mathbb R^N, $

$u(x) = v(x) =0,$ for $|x| =1, $

$u(x), v(x) \rightarrow 0 $ as $|x| \rightarrow +\infty,$

where $1 < p < N $ and $\Delta_p u=$ div $(|\nabla u|^{p-2}\nabla u )$ is the p-Laplacian operator. We consider nonlinearities that are either superlinear or sublinear.

keywords: Elliptic systems exterior domains positive radial solutions p-Laplacian.
CPAA
Positive radial solutions of a nonlinear boundary value problem
Patricio Cerda Leonelo Iturriaga Sebastián Lorca Pedro Ubilla
Communications on Pure & Applied Analysis 2018, 17(5): 1765-1783 doi: 10.3934/cpaa.2018084
In this work we study the following quasilinear elliptic equation:
$\left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\frac{{|x{|^\alpha }\nabla u}}{{{{(a(|x|) + g(u))}^\gamma }}}) = |x{|^\beta }{u^p}}&{{\rm{in}} \ \Omega }\\{u = 0}&{{\rm{on}}\;\;\;\;\partial \Omega }\end{array}} \right.$
where
$ a $
is a positive continuous function,
$ g $
is a nonnegative and nondecreasing continuous function,
$ Ω = B_R $
, is the ball of radius
$ R>0 $
centered at the origin in
$ \mathbb{R} ^N $
,
$N≥3 $
and, the constants
$ α,β∈\mathbb{R} $
,
$ γ∈(0,1) $
and
$ p>1 $
.
We derive a new Liouville type result for a kind of "broken equation". This result together with blow-up techniques, a priori estimates and a fixed-point result of Krasnosel'skii, allow us to ensure the existence of a positive radial solution. In this paper we also obtain a non-existence result, proven through a variation of the Pohozaev identity.
keywords: Quasilinear elliptic equations A priori estimates Liouville theorems fixed point theorems

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