TY - JOUR
T1 - Existence and nonexistence results for critical growth biharmonic elliptic problems
AU - Squassina, Marco
PY - 2003
Y1 - 2003
N2 - 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.
AB - 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.
KW - biharmonic critical problems
KW - biharmonic critical problems
UR - http://hdl.handle.net/10807/91016
U2 - 10.1007/s00526-002-0182-9
DO - 10.1007/s00526-002-0182-9
M3 - Article
SN - 0944-2669
VL - 18
SP - 117
EP - 143
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
ER -