Four-parameter bifurcation for a \(p\)-Laplacian system

Fleckinger, Jacqueline ; Pardo, Rosa ; de Thélin, François

Electronic journal of differential equations, Tome 2001 (2001)

Résumé

We study a four-parameter bifurcation phenomenum arising in a system involving

$\displaylines{ -\Delta_p u = a \phi_p(u)+ b \phi_p(v) + f(a , \phi_p (u), \phi_p (v)) ,\cr -\Delta_p v = c \phi_p(u) + d \phi{p}(v)) + g(d , \phi_p (u), \phi_p (v)), }$

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 $p greater than 1$ and $p \neq 2$, is the $p$-Laplacian operator, and $\phi_{p} (s) =|s|^{p-2} s$ with $p greater than 1$. We assume that $a, b, c, d$ are real parameters. Then we use a bifurcation method to exhibit some nontrivial solutions. The associated eigenvalue problem, with $f=g \equiv 0$, is also studied here.

Détail

Classification : 35J45, 35J55, 35J60, 35J65, 35J30, 35P30
Keywords: p-Laplacian, bifurcation