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

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.

Keywords

• Degenerate nonlinear elliptic second-order equation
• $L^1$-data
• power weights
• Dirichlet problem
• weak solution
• existence and nonexistence of weak solutions