| Popis: |
In this paper, we study the following non-linear system on $mathbb{R}^N$ $$displaylines{ -Delta_pu=a(x)|u|^{p-2}u+b(x)|u|^{alpha}|v|^{eta}v+fquad xin mathbb{R}^Ncr -Delta_qv=c(x)|u|^{alpha}|v|^{eta}u+d(x)|v|^{q-2}v+g quad xin mathbb{R}^Ncr lim_{|x|oinfty}u(x)=lim_{|x|oinfty}v(x)=0,quad u,v>0quad hbox{in }mathbb{R}^Ncr }$$ where $Delta_pu=hbox{m div}| abla u|^{p-2} abla u)$ with $ p$ greater than 1 and $p eq 2$ is the ``p-Laplacian", $alpha,eta$ greater than 0, $p,q$ greater than 1, and $f,g$ are given functions. We obtain necessary and sufficient conditions for having a maximum principle; then we use an approximation method to prove the existence of positive solution for this system. |
