Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ 2 u = u | u | p − 1 with p ∈ ( 1 , ∞ ) has positive solutions on R n if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmet...

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Bibliographic Details
Published in:Journal of Differential Equations 2009-09, Vol.247 (5), p.1479-1504
Main Authors: Lazzo, M., Schmidt, P.G.
Format: Article
Language:English
Subjects:
Biharmonic equation
Dirichlet problem
Entire solutions
Large solutions
Oscillatory behavior
Radial solutions
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Summary:It is well known that the biharmonic equation Δ 2 u = u | u | p − 1 with p ∈ ( 1 , ∞ ) has positive solutions on R n if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on R n in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2009.05.005