Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 39, pp. 1-15.

Title: Existence results for  quasilinear  elliptic systems in $R^N$

Authors: N. M. Stavrakakis (National Technical Univ., Athens, Greece)
 N. B. Zographopoulos (National Technical Univ., Athens, Greece)

Abstract: 
 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
 $\lim_{|x| \to \infty} u(x) =\lim_{|x| \to \infty} u(x)=0$.

Submitted July 16, 1999. Published October 4, 1999.
Math Subject Classifications: 35P30, 35J70, 35B45, 35B65.
Key Words: p-Laplacian; nonlinear eigenvalue problems;
     homogeneous Sobolev spaces; maximum principle; Palais-Smale Condition.