Nonvariational Semilinear Elliptic Systems

In this paper we survey questions regarding the existence of solutions of the Dirichlet problem for systems of semilinear elliptic equations of the type 1 − Δ u = f ( x , u , v , ∇ u , ∇ v ) , − Δ v = g ( x , u , v , ∇ u , ∇ v ) in Ω , $$\displaystyle \begin{aligned} {-}\varDelta u = f(x,u,v,\nabla u,\nabla v), \ -\varDelta v = g(x,u,v,\nabla u,\nabla v) \ \mathrm{in}\ \varOmega, \end{aligned} $$ where Ω is a bounded subset of ℝ N , N ≥ 3 $$ \mathbb {R}^{N}, N\geq 3$$ . The existence of solutions is discussed here using Topological Methods. In order to use this method, the main point is the proof of a priori bounds for the eventual solutions of the systems above. These bounds will be obtained by three different methods, namely Hardy-type inequalities, Moving Planes techniques, and Blow-up. This last method leads to interesting questions about Liouville problems for systems.