Existence and regularity results for a class of equations with logarithmic growth. (September 2015)

• Record Type: Journal Article
• Authors: Passarelli di Napoli, Antonia
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 290
• Page End: 309
• Publication Date: 2015-09
• Subjects: 35B65 -- 35J50 -- 49J25; Existence -- Regularity -- Logarithmic growth; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.05.025
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the existence and the regularity of the solutions of the following Dirichlet problem { − div ( A ( x, ∇ u ) ) = f in Ω u = 0 on ∂ Ω where A ( x, ξ ) is slowly increasing at ∞ as | ξ | → + ∞ . More precisely, we consider the model case A ( x, ξ ) = M ( x ) ξ | ξ | log α ( 1 + | ξ | ), where M ( x ) is a bounded elliptic matrix and α > 0 is a fixed exponent. We will show that if f ∈ L n ( Ω ) then there exists a unique solution u ∈ W 0 1, 1 ( Ω ) ∩ L ∞ ( Ω ), such that ∇ u ∈ L log α L ( Ω ) . Moreover, we prove that there exists an exponent ε > 0 such that | ∇ u | log α + ε ( 1 + | ∇ u | ) ∈ L 1 ( Ω ) . In case the matrix M ( x ) is assumed to be Hölder continuous, we establish a higher differentiability result for the solution in the scale of fractional order Sobolev spaces which yields a higher integrability result in the scale of Lebesgue spaces.