An Existence and Uniqueness of the Weak Solution of the Dirichlet Problem With the Data in Morrey Spaces

Abstract

Let n-2<\lambda<n , f be a function in Morrey spaces L^{1,\lambda}(\Omega) , and the equation
 Lu=f
 u \in W^{1,2}(\Omega)
 be a Dirichlet problem, where \Omega is a bounded open subset of R^{n} , n \ge 3 , L and is a divergent elliptic operator. In this paper, we prove the existence and uniqueness of this Dirichlet problem by directly using the Lax-Milgram Lemma and the weighted estimation in Morrey spaces

An Existence and Uniqueness of the Solution of Semilinear Monotone Elliptic Equation With the Data in Stummel Classes

“…For 𝑛 − 2 < 𝜆 < 𝑛, the existence, uniqueness, and the regularity of the problem (1) obtained by [8,6], but the detail proof for the existence and uniqueness of the problem was given by [10]. Meanwhile, for 0 < 𝜆 < 𝑛 − 2, the problem (1) was studied by [11,6,14]. For the case 𝑓 ∈ 𝑆 ̃2(Ω) and the linear case 𝑔 = 0, the existence, uniqueness, and the regularity of the problem (1) was studied by [8].…”