Existence of positive radial solutions for some nonvariational superlinear elliptic systems

The system under consideration is − Δ u + c u = g ( u , v ) + u p , u = u ( x ) , x ∈ B ⊂ R N , u | ∂ B = 0 , − Δ v + d v = h ( u , v ) + v q , v = v ( x ) , v | ∂ B = 0 , where c , d ⩾ 0 are constants, B is a ball and 1 < p , q < p ⋆ with p ⋆ = ( N + 2 ) / ( N − 2 ) if N ⩾ 3 and p ⋆ = + ∞ if N = 1 , 2 . Among others, it is assumed that g ( 0 , v ) = h ( u , 0 ) = g u ′ ( 0 , v ) = h v ′ ( u , 0 ) = 0 and that g and h are nondecreasing functions in each of their arguments obeying certain growth conditions at infinity. We prove the existence of a radial solution ( u , v ) satisfying u , v > 0 in B.