Abundance of entire solutions to nonlinear elliptic equations by the variational method

We study entire bounded solutions to the equation Δu−u+u3=0 in R2. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in an unified way several types of solutions with various symmetries (radial, breather type, r...

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Bibliographic Details
Published inNonlinear analysis Vol. 190; p. 111590
Main Authors Lerman, L.M., Naryshkin, P.E., Nazarov, A.I.
Format Journal Article
LanguageEnglish
Published Elmsford Elsevier Ltd 01.01.2020
Elsevier BV
Subjects
Elliptic functions
Mathematical analysis
Nonlinear equations
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Summary:We study entire bounded solutions to the equation Δu−u+u3=0 in R2. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in an unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. It is also applicable for more general equations in any dimension.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2019.111590