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Existence and nonexistence results for critical growth biharmonic elliptic equations

by Gazzola, F., Grunau, H.-Ch., Squassina, M..

Series: 2011-02, Preprints

35J65 Nonlinear boundary value problems for linear elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
58E05 Abstract critical point theory (Morse theory, ~Ljusternik-Schnirelman (~Lyusternik-Shnirel'man) theory, etc.)

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.

Critical exponent, best Sobolev constant, semilinear biharmonic problem, Navier boundary condition