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.