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Existence and nonexistence results for critical growth biharmonic elliptic equations

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Abstract.

We prove existence of nontrivial solutions to semilinear fourth order problems at critical growth in some contractible domains which are perturbations of small capacity of domains having nontrivial topology. Compared with the second order case, some difficulties arise which are overcome by a decomposition method with respect to pairs of dual cones. In the case of Navier boundary conditions, further technical problems have to be solved by means of a careful application of concentration compactness lemmas. The required generalization of a Struwe type compactness lemma needs a somehow involved discussion of certain limit procedures. Also nonexistence results for positive solutions in the ball are obtained, extending a result of Pucci and Serrin on so-called critical dimensions to Navier boundary conditions. A Sobolev inequality with optimal constant and remainder term is proved, which is closely related to the critical dimension phenomenon. Here, this inequality serves as a tool in the proof of the existence results and in particular in the discussion of certain relevant energy levels.

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Authors and Affiliations

  1. Dipartimento di Scienze e Tecnologie Avanzate, Corso Borsalino 54, 15100, Alessandria, Italy

    Filippo Gazzola

  2. Fakultät für Mathematik der Universität, Postfach 4120, 39016, Magdeburg, Germany

    Hans-Christoph Grunau

  3. Dipartimento di Matematica e Fisica, Via Musei 41, 25121, Brescia, Italy

    Marco Squassina

Authors
  1. Filippo Gazzola
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  2. Hans-Christoph Grunau
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  3. Marco Squassina
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Correspondence to Filippo Gazzola.

Additional information

Received: 18 April 2002, Accepted: 3 September 2002, Published online: 17 December 2002

Mathematics Subject Classification (2000):

35J65; 35J40, 58E05

The first author was supported by MURST project "Metodi Variazionali ed Equazioni Differenziali non Lineari".

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Gazzola, F., Grunau, HC. & Squassina, M. Existence and nonexistence results for critical growth biharmonic elliptic equations. Cal Var 18, 117–143 (2003). https://doi.org/10.1007/s00526-002-0182-9

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