Multiple solutions for a class of nonhomogeneous elliptic systems with Dirichlet boundary or Neumann boundary

In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ and $N\ge 1$. We exploit the method which is based on [6]. This method let us obtain the concrete open interval about the parameter $\lambda$. Since the quasilinear term depends on $u$ and $\mathrm{\nabla }u$, it is necessary for our proofs to use the theory of monotone operators and the skill of adding one dimension to space.