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Elliptic problems involving the 1–Laplacian and a singular lower order term
Author(s) -
De Cicco V.,
Giachetti D.,
Segura de León S.
Publication year - 2019
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12172
Subject(s) - uniqueness , mathematics , bounded function , lipschitz continuity , pure mathematics , term (time) , divergence (linguistics) , order (exchange) , dirichlet problem , limit (mathematics) , type (biology) , operator (biology) , laplace operator , p laplacian , boundary (topology) , open set , mathematical analysis , weak solution , boundary value problem , physics , finance , quantum mechanics , ecology , linguistics , philosophy , biochemistry , chemistry , repressor , gene , transcription factor , economics , biology
This paper is concerned with the Dirichlet problem for an equation involving the 1–Laplacian operatorΔ 1 u : = div( D u | D u | )and having a singular term of the typef ( x ) u γ . Here f ∈ L N ( Ω )is nonnegative, 0 < γ ⩽ 1 and Ω is an open bounded set with Lipschitz‐continuous boundary. We prove an existence result for a concept of solution conveniently defined. The solution is obtained as limit of solutions to p ‐Laplacian type problems. Moreover, when f ( x ) > 0 almost everywhere, the solution satisfies those features that might be expected as well as a uniqueness result. We also give explicit one–dimensional examples that show that, in general, uniqueness does not hold. We remark that the Anzellotti theory of L ∞ –divergence–measure vector fields must be extended to deal with this equation.