Geodesic
On a sharp condition for the existence of weak solutions to the Dirichlet problem for degenerate nonlinear elliptic equations with power weights and $L^1$-data
Electronic Journal of Differential Equations, Tome 2015 (2015).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: In this article, we establish a sharp condition for the existence of weak solutions to the Dirichlet problem for degenerate nonlinear elliptic second-order equations with $L^1$-data in a bounded open set $\Omega$ of $\mathbb{R}^n$ with $n\geq 2$. We assume that $\Omega$ contains the origin and assume that the growth and coercivity conditions on coefficients of the equations involve the weighted function $\mu(x)=|x|^\alpha$, where $\alpha\in (0,1]$, and a parameter $p\in (1,n)$. We prove that if $p>2-(1-\alpha)/n$, then the Dirichlet problem has weak solutions for every $L^1$-right-hand side. On the other hand, we find that if $p\leq 2-(1-\alpha)/n$, then there exists an $L^1$-datum such that the corresponding Dirichlet problem does not have weak solutions.
Classification : 35J25, 35J60, 35J70, 35R05
Keywords: degenerate nonlinear elliptic second-order equation, $L^1$-data, power weights, Dirichlet problem, weak solution, existence and nonexistence of weak solutions 
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     author = {Kovalevsky, Alexander A. and Nicolosi, Francesco},
     title = {On a sharp condition for the existence of weak solutions to the {Dirichlet} problem for degenerate nonlinear elliptic equations with power weights and $L^1$-data},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2015},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a2/}
}
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Kovalevsky, Alexander A.; Nicolosi, Francesco. On a sharp condition for the existence of weak solutions to the Dirichlet problem for degenerate nonlinear elliptic equations with power weights and $L^1$-data. Electronic Journal of Differential Equations, Tome 2015 (2015). http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a2/