Regularity of the extremal solutions associated to elliptic systems

We examine the two elliptic systems given by(G)λ,γ−Δu=λf′(u)g(v),−Δv=γf(u)g′(v) inΩ, and(H)λ,γ−Δu=λf(u)g′(v),−Δv=γf′(u)g(v) inΩ, with zero Dirichlet boundary conditions and where λ, γ are positive parameters. We show that for general nonlinearities f and g   the extremal solutions associated with (G)λ,γ(G)λ,γ are bounded, provided Ω   is a convex domain in RNRN where N≤3N≤3. In the case of a radial domain, we show the extremal solutions are bounded provided N<10N<10. The extremal solutions associated with (H)λ,γ(H)λ,γ are bounded in the case where f   is a general nonlinearity, g(v)=(v+1)qg(v)=(v+1)q for 1