Electronic Journal of Differential Equations,
Vol. 2015 (2015), No. 52, pp. 1-12.
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
Authors: Alexander A. Kovalevsky (Krasovsky Inst. of Math and Mech., Ekaterinburg, Russia)
Francesco Nicolosi (Univ. of Catania, Italy)
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.