Optimal Liouville-type theorems for polyharmonic elliptic gradient systems and applications

Abstract

In this paper, we investigate the following polyharmonic systems (d≥1) in RN,{(−Δ)du1=f1(u1,u2),(−Δ)du2=f2(u1,u2). By using the method of Rellich-Pohozaev type identities, Sobolev and interpolation inequalities on SN−1 and feedback and measure arguments, we prove Liouville-type theorems for nonnegative classical solutions under an optimal Sobolev growth condition on f1 and f2. Furthermore, we prove priori bounds, decay and singularity estimates of solutions. This system can be applied to noncooperative Schrödinger system with polyharmonic operator which arises from nonlinear optics and Bose-Einstein condensates.