Abstract
We study nonlinear measure data elliptic problems involving the operator of generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. Approximable and renormalized solutions are proven to exist and coincide for arbitrary measure datum and to be unique when for a class of data being diffuse with respect to a relevant nonstandard capacity. A capacitary characterization of diffuse measures is provided.
Highlights
Our objective is to study existence and uniqueness of two kinds of very weak solutions to nonlinear measure data problem
Very weak solutions to measure-data problems of the form (1) are already studied in depth in the classical setting of Sobolev spaces, that is when the growth growth of the leading part of the operator is governed by a power function with the celebrated special case of p-Laplacian ∆pu = div (|∇u|p−2∇u)
To give a flavour let us mention e.g. [12, 16, 17, 34, 35], where the existence is provided for various notions of very weak solutions for L1 or measure data
Summary
The existence of renormalized solutions to general measure data problem and uniqueness for diffuse measures is new even in the reflexive Orlicz case. Existence of very weak solutions and uniqueness in the case of diffuse measures is studied in the variable exponent setting in [67, 55]. The special case of this result is the classical measure characterization [16]: if p > 1, μp ∈ b(Ω) does not charge the sets of the Sobolev p-capacity zero if and only if μp ∈ L1(Ω) + W −1,p′(Ω), i.e. there exist f ∈ L1(Ω) and G ∈ (Lp′ (Ω))n, such that μp = f − div G in the sense of distributions. Having [35] and Remark 2 on measure decomposition (to parts being absolutely continuous and singular with respect to generalized capacity) we consider renormalized solutions according to the following definition.
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