On the existence of radial positive entire solutions for polyharmonic systems (Q860955)

The nonlinear system \(\Delta (| \Delta^n u_j| ^{p_j-2}\Delta^n u_j)=f_j(| x| ,u_1, u_2,| \nabla u_1| ,| \nabla u_2| )\), \(j=1,2\), is studied in \(\mathbb R^2\). Here \(n\in \mathbb{N}\) and \(p_1,p_2>1\) are constants. Some sufficient conditions are obtained for the existence of infinitely many radial positive entire solutions of the system such that \(u_j(x)| x| ^{-2n}(\log | x| )^{ 1/(1-p_j)} \to A_j\), \(| \nabla u_j(x)| | x| ^{1-2n}(\log | x| )^{1/(1-p_j)}\to B_j\) as \(| x| \to \infty \), where \(A_j>0\), \(B_j>0\) are constants depending only on \(u_j\).