Constant Sign and Sign Changing Solutions for Systems of Quasilinear Elliptic Equations

We establish the existence of two opposite constant sign solutions for a general noncoercive quasilinear elliptic system with homogeneous Dirichlet boundary conditions. In the case where the system has a variational structure, by strengthening the hypotheses, we obtain a third nontrivial solution which is sign changing in the sense that one cannot have both components of the new solution of the same constant sign. Our approach relies on a suitable method of sub-supersolutions combined with truncation and variational arguments that does not require a subcritical growth condition.