Existence of positive solutions for $p$--Laplacian problems with weights
Friedemann Brock Leonelo Iturriaga Justino Sánchez Pedro Ubilla
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
Positive radial solutions for some quasilinear elliptic systems in exterior domains
João Marcos do Ó Sebastián Lorca Justino Sánchez Pedro Ubilla
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.

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