MULTIPLICITY OF SOLUTIONS FOR EQUATIONS INVOLVING A NONLOCAL TERM AND THE BIHARMONIC OPERATOR

In this work we study the existence and multiplicity result of solutions to the equation Delta(2)u - M(integral(Omega)|del u|(2)dx)Delta u = lambda|u|(q) (2)u + |u|(2)***u in Omega, u = Delta u = 0 on partial derivative Omega, where Omega is a bounded smooth domain of R-N, N >= 5, 1 < q < 2 or 2 < q < 2**, M : R+ -> R+ is a continuous function. Since there is a competition between the function M and the critical exponent, we need to make a truncation on the function M. This truncation allows to define an auxiliary problem. We show that, for lambda large, exists one solution and for lambda small there are in finitely many solutions for the auxiliary problem. Here we use arguments due to Brezis-Niremberg [12] to show the existence result and genus theory due to Krasnolselskii [29] to show the multiplicity result. Using the size of lambda, we show that each solution of the auxiliary problem is a solution of the original problem.