On partially free boundary solutions for elliptic problems with non-Lipschitz nonlinearities

We show that the elliptic equation with a non-Lipschitz right-hand side, −Δu=λ|u|β−1u−|u|α−1u with λ>0 and 0<α<β<1, considered on a smooth star-shaped domain Ω subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf’s maximum p...

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Bibliographic Details
Published inApplied mathematics letters Vol. 95; pp. 23 - 28
Main Authors Bobkov, V., Drábek, P., Ilyasov, Y.
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.09.2019
Subjects
Compact support solutions
Compactons
Free boundary solutions
Non-Lipschitz nonlinearity
Non-Lipschitz nonlinearity
Free boundary solutions
Compactons
Compact support solutions
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ISSN0893-9659
1873-5452
DOI10.1016/j.aml.2019.03.019

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Summary:We show that the elliptic equation with a non-Lipschitz right-hand side, −Δu=λ|u|β−1u−|u|α−1u with λ>0 and 0<α<β<1, considered on a smooth star-shaped domain Ω subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf’s maximum principle only on a nonempty subset Γ of the boundary ∂Ω such that Γ≠∂Ω.
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2019.03.019