Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 06, pp. 1-15.
Title: Four-parameter bifurcation for a p-Laplacian system
Authors: Jacqueline Fleckinger (Univ. Toulouse 1, France)
Rosa Pardo (Univ. Complutense de Madrid, Spain)
Francois de Thelin (Univ. Toulouse 3, France)
Abstract:
We study a four-parameter bifurcation phenomenum arising in a
system involving $p$-Laplacians:
$$\displaylines{
-\Delta_p u = a \phi_p(u)+ b \phi_p(v) + f(a , \phi_p (u), \phi_p (v)) ,\cr
-\Delta_p v = c \phi_p(u) + d \phi{p}(v)) + g(d , \phi_p (u), \phi_p (v)),
}$$
with $u=v=0$ on the boundary of a bounded and sufficiently smooth domain in
$\mathbb{R}^N$;
here $\Delta_{p}u = {\rm div} (| \nabla u|^{p-2} \nabla u)$,
with $p>1$ and $p \neq 2$, is the $p$-Laplacian operator, and
$\phi_{p} (s) =|s|^{p-2} s$ with $p>1$.
We assume that $a, b, c, d$ are real parameters, and
use a bifurcation method to exhibit some nontrivial solutions.
The associated eigenvalue problem, with $f=g \equiv 0$, is also studied here.
Submitted June 29, 2000. Published January 9, 2001.
Math Subject Classifications: 35J45, 35J55, 35J60, 35J65, 35J30, 35P30.
Key Words: p-Laplacian; bifurcation.