Stability of positive solutions to biharmonic equations on Heisenberg group

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.