Electron. J. Diff. Equ., Vol. 2015 (2015), No. 52, pp. 1-12.

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

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 $L^1$-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.

Submitted August 5, 2014. Published February 25, 2015.
Math Subject Classifications: 35J25, 35J60, 35J70, 35R05.
Key Words: 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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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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