Four-parameter bifurcation for a p-Laplacian system

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 $Delta_{p}u = { m div} (| abla u|^{p-2} abla u)$, with $p>1$ and $p eq 2$, 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.

Keywords

• p-Laplacian
• bifurcation.