02-11
Existence and nonexistence results for critical growth biharmonic elliptic equations
by Gazzola, F.; Grunau, H.-Ch.; Squassina, M.
Preprint series: 02-11, Preprints
The paper is published: Calc. Var. PDE 18 , 117-143 (2003).
- MSC:
- 35J65 Nonlinear boundary value problems for linear elliptic PDE; boundary value problems for nonlinear elliptic PDE
- 35J40 Boundary value problems for higher-order, elliptic equations
- 58E05 Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirelman) theory, etc.)
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. Also the required generalization of a Struwe type compactness lemma needs a somehow involved discussion of certain limit procedures. A Sobolev inequality with optimal constant and remainder term is proved, which may be of interest not only as a technical tool. Finally, 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.
Keywords: Critical exponent, best Sobolev constant, semilinear biharmonic problem, Navier boundary condition
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