In this paper, we characterise the structure of the eigencone for the Finsler Laplacian corresponding to the first Dirichlet eigenvalue on a compact Finsler manifold with a smooth boundary.

We consider the quasilinear elliptic variational system

\begin{alignat*} {2} -\Delta_pu\amp=\lambda F_u(x,u,v)+\mu H_u(x,u,v) \quad\amp \amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)+\mu H_v(x,u,v) \amp \amp \text{in }\varOmega, \\ u\amp=v=0 \amp \amp \text{on }\partial\varOmega, \end{alignat*}

where $\varOmega$ is a strip-like domain and $\lambda$ and $\mu$ are positive parameters. Using a recent two-local-minima theorem and the principle of symmetric criticality, existence and multiplicity are proved under suitable conditions on $F$.

In this paper we study the multiplicity of solutions of the quasilinear elliptic system

\begin{equation} \left. \begin{aligned} -\Delta_pu\amp=\lambda F_u(x,u,v)\amp\amp\text{in }\varOmega, \\ -\Delta_qv\amp=\lambda F_v(x,u,v)\amp\amp\text{in }\varOmega, \\ u=v\amp=0\amp\amp\text{on }\partial\varOmega, \end{aligned} \right\} \end{equation} \tag{S$_\lambda$}

where $\varOmega$ is a strip-like domain and $\lambda>0$ is a parameter. Under some growth conditions on $F$, we guarantee the existence of an open interval $\varLambda\subset(0,\infty)$ such that for every $\lambda\in\varLambda$, the system (S$_\lambda$) has at least two distinct, non-trivial solutions. The proof is based on an abstract critical-point result of Ricceri and on the principle of symmetric criticality.