Regularity of the extremal solutions associated to elliptic systems.

We examine the two elliptic systems given by ( G ) λ , γ − Δ u = λ f ′ ( u ) g ( v ) , − Δ v = γ f ( u ) g ′ ( v ) in Ω , and ( H ) λ , γ − Δ u = λ f ( u ) g ′ ( v ) , − Δ v = γ f ′ ( u ) g ( v ) in Ω , 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 are bounded, provided Ω is a convex domain in R N where N ≤ 3 . In the case of a radial domain, we show the extremal solutions are bounded provided N < 10 . The extremal solutions associated with ( H ) λ , γ are bounded in the case where f is a general nonlinearity, g ( v ) = ( v + 1 ) q for 1 < q < ∞ and when Ω is a bounded convex domain in R N for N ≤ 3 . Certain regularity results are also obtained in higher dimensions for ( G ) λ , γ and ( H ) λ , γ for the case of explicit nonlinearities of the form f ( u ) = ( u + 1 ) p and g ( v ) = ( v + 1 ) q . [ABSTRACT FROM AUTHOR]