On the Existence of Many Solutions for a Class of Superlinear Elliptic Systems

We seek multiple solutions to systems of semilinear elliptic equations of the type−ΔU(x)= H(x; U(x)) x ΩU(x)=0 x ∂Ω,where Ω N (N 1) is a bounded regular domain and U=(u1, u2): Ω→ 2. For a class of potentials where the variables are strongly coupled as in the model H(x; u1, u2)=u1α1 u2α2 with αi>1, we provide the existence of nine nontrivial solutions characterized by sign properties of each component.