Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

Abstract

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form $$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$ L u : = ∑ i , j = 1 n a ij ( x ) u x i x j = ∇ · f , for almost all x ∈ Ω where the coefficients $$a_{ij}$$ a ij belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces $$\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$$ LM ~ { x 0 } p , φ . Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.

Ground State Solution for an Autonomous Nonlinear Schrödinger System

“…There are some recent studies on the ground state solutions for other types of Schrödinger equations or systems, for example, [6,11]. Moreover, in the bounded domain, the existence and the regularity of solutions to differential problems have been widely investigated by using tools of harmonic and real analysis and variational methods, for example, [12][13][14].…”

Multi-sublinear operators and their commutators on product generalized mixed Morrey spaces

“…One can see [1,5,23] and some related papers for the boundedness of some classical operators on M p q (R n ). Recently, Guliyev et al [4,12,11,14] extended M p q (R n ) to the generalized Morrey spaces M ϕ q (R n ) and studied the regularity of solutions of elliptic equations in divergence form in the generalized Morrey spaces.…”

Boundedness criterion for sublinear operators and commutators on generalized mixed Morrey spaces

“…Since M α f (x) I α |f |(x), 0 < α < n, the fractional maximal operator M α is also bounded in M p q (R n ). In recent years, the classical Morrey spaces M p q (R n ) was extended to the generalized Morrey space by Guliyev et al [3,9,11,12].…”