Bifurcation results for quasilinear elliptic systems

Abstract

We prove certain bifurcation results for the quasilinear elliptic system \begin{align*} & -\Delta_{p}u = \lambda\, a(x)\, |u|^{p-2}u+\lambda\, b(x)\, |u|^{\alpha}\, |v|^{\beta}\, v +f(x,\lambda,u,v), \\ & -\Delta_{q}v = \lambda\, d(x)\, |v|^{q-2}v+\lambda\, b(x)\, |u|^{\alpha}\, |v|^{\beta}\, u +g(x,\lambda,u,v), \end{align*} defined on an arbitrary domain (bounded or unbounded) of $\mathbb{R}^N$, where the functions $a$, $d$, $f$ and $g$ may change sign. To this end we establish the isolation of the principal eigenvalue of the corresponding unperturbed system and apply topological degree theory.