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.

Gazzola, F., Grunau, H. C., Squassina, M., Existence and nonexistence results for critical growth biharmonic elliptic problems, <<CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS>>, 2003; 18 (N/A): 117-143 [http://hdl.handle.net/10807/91016]

Existence and nonexistence results for critical growth biharmonic elliptic problems

Squassina, Marco
Ultimo
2003

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.
Inglese
Gazzola, F., Grunau, H. C., Squassina, M., Existence and nonexistence results for critical growth biharmonic elliptic problems, <<CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS>>, 2003; 18 (N/A): 117-143 [http://hdl.handle.net/10807/91016]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/91016
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