Alexander A. Kovalevsky, Francesco Nicolosi
Abstract:
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
-data in a bounded
open set
of
with
.
We assume that
contains the origin and assume that
the growth and coercivity conditions on coefficients of the equations
involve the weighted function
,
where
,
and a parameter
.
We prove that if
,
then the Dirichlet problem
has weak solutions for every
-right-hand side.
On the other hand, we find that if
,
then there exists an
-datum
such that the corresponding Dirichlet problem
does not have weak solutions.
Submitted August 5, 2014. Published February 25, 2015.
Math Subject Classifications: 35J25, 35J60, 35J70, 35R05.
Key Words: Degenerate nonlinear elliptic second-order equation;
-data;
power weights; Dirichlet problem; weak solution;
existence and nonexistence of weak solutions.
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Alexander A. Kovalevsky Department of Equations of Mathematical Physics Krasovsky Institute of Mathematics and Mechanics Ural Branch of Russian Academy of Sciences Ekaterinburg, Russia email: alexkvl71@mail.ru | |
Francesco Nicolosi Department of Mathematics and Informatics University of Catania Catania, Italy email: fnicolosi@dmi.unict.it |
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