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
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
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发表时间：
2015
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通讯作者：
A. Kovalevsky;F. Nicolosi
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作者：
A. Kovalevsky;F. 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 L1-data in a bounded open set Ω of Rn with n > 2. We assume that Ω contains the origin and assume that the growth and coercivity conditions on coefficients of the equations involve the weighted function μ(x) = |x|α, where α ∈ (0, 1], and a parameter p ∈ (1, n). We prove that if p > 2 − (1 − α)/n, then the Dirichlet problem has weak solutions for every L1-right-hand side. On the other hand, we find that if p 6 2−(1−α)/n, then there exists an L1-datum such that the corresponding Dirichlet problem does not have weak solutions.
