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Title: Extremal solutions of strongly coupled nonlinear elliptic systems and L∞-boundedness.
Authors: Costa, Felipe
Souza, Gil Fidelix de
Montenegro, Marcos
Keywords: Positive solutions
Issue Date: 2022
Citation: COSTA, F.; SOUZA, G. F. de.; MONTENEGRO, M. Extremal solutions of strongly coupled nonlinear elliptic systems and L∞-boundedness. Journal of Mathematical Analysis and Applications, v. 513, n. 2, artigo 126225, 2022. Disponível em: <>. Acesso em: 06 jul. 2022.
Abstract: The paper concerns positive solutions for the Dirichlet problem −Lu = ΛF(x, u) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n ≥ 2, u = (u1, ..., um) : Ω → Rm, m ≥ 1, Lu = (L1u1, ..., Lmum), where each Li denotes a uniformly elliptic linear operator of second order in nondivergence form in Ω, Λ = (λ1, ..., λm) ∈ Rm, F = (f1, ..., fm) : Ω × Rm → Rm and ΛF(x, u) = (λ1f1(x, u), ..., λmfm(x, u)). For a general class of maps F we prove that there exists a hypersurface Λ∗ in Rm + := (0, ∞)m such that tuples Λ ∈ Rm + below Λ∗ correspond to minimal positive strong solutions of the above system. Stability of these solutions is also discussed. Already for tuples above Λ∗, there is no nonnegative strong solution. The shape of the hypersurface Λ∗ depends on growth on u of the nonlinearity F in a sense to be specified. When Λ ∈ Λ∗ and the coefficients of each operator Li are slightly smooth, the problem admits a unique minimal nonnegative weak solution, called extremal solution. Furthermore, when F depends only on u and all Li are Laplace operators, we investigate the L∞ regularity of this solution for any m ≥ 1 in dimensions 2 ≤ n ≤ 9 for balls and n = 2 and n = 3 for convex domains.
ISSN: 0022-247X
Appears in Collections:DEMAT - Artigos publicados em periódicos

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