On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|∇   u   i| p –2 ∇   u   i) + λk i (| x |) f i ( u 1 , …, u n ) = 0, p > 1, R 1 < | x | < R 2 , u i ( x ) = 0, on | x | = R 1 and R 2 , i = 1, …, n , x ∈ ℝ N , where k i and f i , i = 1, …, n , are continuous and nonnegative functions. Let u = ( u 1 , …, u n ), φ ( t ) = | t | p –2 t , f i 0 = lim ‖ u ‖→0 (( f i ( u ))/( φ (‖ u ‖))), f i ∞ = lim ‖ u ‖→∞ (( f i ( u ))/( φ (‖ u ‖))), i = 1, …, n , f = ( f 1 , …, f n ), f 0 = ∑ ni =1 f i 0 and f ∞ = ∑ ni =1 f i ∞ . We prove that either f 0 = 0 and f ∞ = ∞ (superlinear), or f 0 = ∞and f ∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if f i ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n , then either f 0 = f ∞ = 0, or f 0 = f ∞ = ∞, guarantee multiplicity for sufficiently large, or small λ , respectively. On the other hand, either f 0 and f ∞ > 0, or f 0 and f ∞ < ∞ imply nonexistence for sufficiently large, or small λ , respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)