Regularity of the extremal solutions associated to elliptic systems

We examine the two elliptic systems given by [(G)_{λ,γ} \quad -Δu = λf'(u) g(v), \quad -Δv = γf(u) g'(v) \quad in $Ω$,] and [(H)_{λ,γ} \quad -Δu = λf(u) g'(v), \quad -Δv = γf'(u) g(v) \quad in $Ω$},] with zero Dirichlet boundary conditions and where $λ,γ$ are positive parameters. We show that for arbitrary nonlinearities $f$ and $g$ that the extremal solutions associated with $(G)_{λ,γ}$ are bounded provided $Ω$ is a convex domain in $\mathbb R^N$ where $N \le 3$. 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 arbitrary, $g(v)=(v+1)^q$ where $1 <q<\infty$ and where $Ω$ is a bounded convex domain in $\mathbb R^N$, $N \le 3$. 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 $g(v)=(v+1)^q$.

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