Abstract:
In this note, we establish the existence of a positive solution and
its semi-stability to the following class of biharmonic problems with logistictype
nonlinearities
(0.1) 2
Hnu = a( )u − f( , u) in
u|@
= 0 = Hnu|@
,
where
Hn is an open, smooth and bounded subset of Heisenberg group
Hn. We establish the existence of a solution by Schauder’s fixed point theorem
and then with the aid of strong maximum principle, we obtain the positivity
of the solution. We also show that the principal eigenvalue of the linearized
equation associated with (0.1) is non-negative and hence the solution u of (0.1)
is semi-stable. This is shown by testing the equation under consideration with
a suitable test function.