Positive radially symmetric ground states to a bi-harmonic problem

Abstract

In this paper, we establish the existence of positive solutions to the semilinear problem for bi-harmonic operator $$ (-\Delta)^2u+u=|u|^{p-1}u, \ \ \text{ in }\ R^n. $$ One key tool is to use the Schwartz symmetrization method, in particular the Riesz inequality to this problem. The other tool is the generalized Ni-Strauss Lemma to radially symmetric functions in higher order Sobolev spaces.