Nonradial solutions of nonlinear scalar field equations

We prove new results concerning the nonlinear scalar field equation − Δ u = g ( u ) in R N , N ⩾ 3 , u ∈ H 1 ( R N ) with a nonlinearity g satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any N ⩾ 4 minimizing the energy functi...

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Bibliographic Details
Published inNonlinearity Vol. 33; no. 12; pp. 6349 - 6380
Main Author Mederski, Jarosław
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.12.2020
Subjects
concentration compactness
critical point theory
nonlinear scalar field equations
nonradial solutions
Pohozaev manifold
profile decomposition
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ISSN0951-7715
1361-6544
DOI10.1088/1361-6544/aba889

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Summary:We prove new results concerning the nonlinear scalar field equation − Δ u = g ( u ) in R N , N ⩾ 3 , u ∈ H 1 ( R N ) with a nonlinearity g satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any N ⩾ 4 minimizing the energy functional on the Pohozaev constraint in a subspace of H 1 ( R N ) consisting of nonradial functions. If in addition N ≠ 5, then there are infinitely many nonradial solutions. These solutions are sign-changing. The results give a positive answer to a question posed by Berestycki and Lions in [5, 6]. Moreover, we build a critical point theory on a topological manifold, which enables us to solve the above equation as well as to treat new elliptic problems.
Bibliography:NON-103763.R2
London Mathematical Society
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/aba889