A remark on the radial and nonradial solutions for an elliptic system with a quadratic gradient term

Abstract

Recently, the authors Yang and Zhang in [Y. Yang, X. Zhang, Entire blow-up solutions of semilinear elliptic system with quadratic gradient term, Abstract and Applied Analysis, 2012] have been obtained sufficiently conditions for the existence and non-existence of radially symmetric solutions for the system of equations Δui+∣∇ui∣2=ai(x)fi(u1,…,ud) on RN (N⩾3) with i=1,…,d and d∈{1,2,3,4,…}. Our aim in this paper is to give a necessary and a sufficient condition, but for the existence of large (or explosive) solutions for the same class of systems. Moreover, we intend to obtain a non-radial solution for the considered problem. We also point out that our framework in the obtained results improve and complete some of the results obtained by Yang and Zhang.