Alexander A. Kovalevsky, Francesco Nicolosi
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.
Show me the PDF file (256 KB), TEX file, and other files for this article.
| Alexander A. Kovalevsky |
Department of Equations of Mathematical Physics
Krasovsky Institute of Mathematics and Mechanics
Ural Branch of Russian Academy of Sciences
| Francesco Nicolosi |
Department of Mathematics and Informatics
University of Catania
Return to the EJDE web page