Regularity of the extremal solutions associated to elliptic systems

We examine the elliptic system given by \begin{eqnarray*} \qquad \left{ \begin{array}{lcl} -\Delta u =\lambda f(v) \quad \mbox{ in } \Omega -\Delta v =\gamma f(u) \quad \mbox{ in } \Omega, u=v =0, \quad \mbox{ on } \pOm \end{array}\right. \end{eqnarray*} where $\lambda ,\gamma$ are positive parameters, $\mathrm{\Omega }$ is a smooth bounded domain in ${\text{IR}}^N$ and $f$ is a $C^2$ positive, nondecreasing and convex function in $[0,\mathrm{\infty })$ such that $\frac{f(t)}{t}\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$. Assuming $$0<{\tau }_{-}:=\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{f(t)f\mathrm{"}(t)}{f^{\mathrm{\prime }}(t{)}^2}\le {\tau }_{+}:=\underset{t\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{f(t)f\mathrm{"}(t)}{f^{\mathrm{\prime }}(t{)}^2}\le 2,$$ we show that the extremal solution $(u^{\ast },v^{\ast })$ associated to the above system is smooth provided\ $N<\frac{2{\alpha }_{\ast }(2-{\tau }_{+})+2{\tau }_{+}}{{\tau }_{+}}\max \{1,{\tau }_{+}\}$, where ${\alpha }_{\ast }>1$ denotes the largest root of the $2^{nd}$ order polynomial $$P_f(\alpha ,{\tau }_{-},{\tau }_{+}):=(2-{\tau }_{-}{)}^2{\alpha }^2-4(2-{\tau }_{+})\alpha +4(1-{\tau }_{+})\mathrm{.}$$ As a consequences, $u^, v^\in L^\infty(\Omega)$for$N<5$\mathrm{.}Moreover,if$\tau_{-}=\tau_{+}$,then$u^, v^\in L^\infty(\Omega)$for$N<10$.