Existence results for quasilinear elliptic systems in RN - N. M. Stavrakakis (Electronic Journal of Differential Equations, 1999)
We prove existence results for the quasilinear elliptic system $$ displaylines{ -Delta_{p}u = lambda a(x)|u|^{gamma-2}u +lambda b(x) |u|^{alpha -1}|v|^{Beta +1}u, cr -Delta_{q}v = lambda d(x)|v|^{delta-2}v +lambda b(x)|u|^{alpha +1}|v|^{Beta -1}v,, cr }$$ where $gamma$ and $delta$ may reach the critical Sobolev exponents, and the coefficient functions $a$, $b$, and $d$ may change sign. For the unperturbed system ($a=0$, $b=0$), we establish the existence and simplicity of a positive principal eigenvalue, under the assumption that $u(x)>0$, $v(x)>0$, and $lim_{|x| o infty} u(x) =lim_{|x| o infty} u(x)=0$.