Entire Blow-Up Solutions of Semilinear Elliptic Systems with Quadratic Gradient Terms

Abstract

We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms, $\mathrm{\Delta }{u}_i+|\nabla u_i{|}^2=p_i(|x|)f_i(u_1,u_2,\dots ,u_d)$ for $i=\mathrm{1,2},\dots ,d$ on $R^N,N\ge 3$ and $d\in \{\mathrm{1,2},3,\dots \}$. We establish the conditions on $p_i$ that ensure the existence of nonnegative radial solutions blowing up at infinity and also the conditions for bounded solutions on the entire space. The condition on $f_i$ is simple and different to the Keller-Osserman condition.