Asymptotic Uniqueness for a Biharmonic Equation with Nearly Critical Growth on Symmetric Convex Domains

Abstract

We consider a biharmonic equation with the nearly critical Sobolev exponent under the Navier boundary condition on a smooth bounded, strictly convex domain of dimension N ≥ 5, which is symmetric with respect to the coordinate hyperplanes.
We prove that the number of positive solutions of the above problem is exactly one when the nonlinear exponent is subcritical and sufficiently near to the critical exponent. Furthermore, this unique solution is nondegenerate in the sense that the associated linearized problem admits only the trivial solution.