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

**Series:** 2011-02, Preprints

- MSC:
- 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.)

**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