Existence of solutions of nonlinear elliptic equations with measure data in Musielak-Orlicz spaces

Abstract

A second-order quasilinear elliptic equation with a measure of special form on the right-hand side is considered. Restrictions on the structure of the equation are imposed in terms of a generalized -function such that the conjugate function obeys the -condition and the corresponding Musielak-Orlicz space is not necessarily reflexive. In an arbitrary domain satisfying the segment property, the existence of an entropy solution of the Dirichlet problem is proved. It is established that this solution is renormalized. Bibliography: 29 titles.

Entropy solution for an equation with measure-valued potential in a hyperbolic space

“…In [9] the Dirichlet problem − div a(x, u, ∇u) + M ′ (x, u) + b(x, u, ∇u) = σ, u| ∂Ω = 0, was considered in an unbounded domain, where the growth of the functions a and b is determined by the generalized N -function M (x, u) and the bounded Radon measure σ is insignificantly different from a function in L 1 (Ω). It was assumed that the conjugate function M (x, u) satisfies the ∆ 2 -condition and b(x, u, ∇u)u ⩾ 0.…”

“…The reader can find a more comprehensive survey of results on entropy and renormalized solutions in [9].…”

“…It is known that the space C ∞ 0 (R n ) can be completed with respect to the norms |∇u| p dx 1/p and (|u| p +|∇u| p ) dx 1/p alike, and in the latter case this yields the more 'narrow' space W 1 p (R n ) ⊂ H 1 p (R n ). Usually authors take the second way (for instance, see [6] and [9]). In our paper we use the space H 1 M (R n ) obtained on the first way.…”