Very singular solutions for linear Dirichlet problems with singular convection terms. (May 2020)

• Record Type: Journal Article
• Authors: Boccardo, Lucio; Orsina, Luigi
• Is Part Of: Nonlinear analysis. Volume 194(2020)
• Publication Date: 2020-05
• Subjects: Convection terms -- Drift terms -- Singular solutions; 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.2019.01.027
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the existence of distributional solutions for the boundary value problems (1.1) and (1.2) if E does not belong to L N, namely | E | ≤ | A | | x |, A ∈ R . The size of A plays an important role: if α ( N − 2 ) ≤ | A | < α ( N − 1 ), we prove that if f ∈ L 1 ( Ω ) there exists a distributional solution u ∈ W 0 1, q ( Ω ), for every q < N α | A | + α < N N − 1, of (1.1) (the case | A | < α ( N − 2 ) is studied in Boccardo (2015)). We then use this result to prove the existence of a bounded weak solution ψ of (1.2) if g ( x ) ∈ L m ( Ω ), m > N α N α − | A | ≥ N 2 .