We consider the semilinear elliptic equation -Lu = f(u) in a general smooth bounded domain ΩRn\documentclass[12pt]{minimal}
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\begin{document}${\Omega } \subset R^{n}$\end{document} with zero Dirichlet boundary condition, where L is a uniformly elliptic operator and f is a C2 positive, nondecreasing and convex function in [0,∞)\documentclass[12pt]{minimal}
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\begin{document}$[0,\infty )$\end{document} such that f(t)t→∞\documentclass[12pt]{minimal}
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\begin{document}$\frac {f(t)}{t}\rightarrow \infty $\end{document} as t→∞\documentclass[12pt]{minimal}
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\begin{document}$t\rightarrow \infty $\end{document}. We prove that if u is a positive semistable solution then for every 0 ≤ β < 1 we have
f(u)∫0uf(t)f''(t)e2β∫0tf''(s)f(s)dsdtL1(O)≤Cβ<∞,\documentclass[12pt]{minimal}
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\begin{document}$$\left|\left|f(u){{\int}_{0}^{u}}f(t)f^{\prime\prime}(t)~e^{2{\beta{\int}_{0}^{t}}\sqrt{\frac{f^{\prime\prime}(s)}{f(s)}}ds}~dt \right|\right|_{L^{1}({\Omega})}\leq C_{\beta}<\infty, $$\end{document}
where Cβ is a constant independent of u. As we shall see, a large number of results in the literature concerning a priori bounds are immediate consequences of this estimate. In particular, among other results, we establish a priori L∞\documentclass[12pt]{minimal}
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\begin{document}$L^{\infty }$\end{document} bound in dimensions n ≤ 9, under the extra assumption that lim supt→∞f(t)f''(t)f'(t)2<29-214≅1.318\documentclass[12pt]{minimal}
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\begin{document}$\limsup _{t\rightarrow \infty }\frac {f(t)f^{\prime \prime }(t)}{f^{\prime }(t)^{2}}< \frac {2}{9-2\sqrt {14}}\cong 1.318$\end{document}. Also, we establish a priori L∞\documentclass[12pt]{minimal}
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\begin{document}$L^{\infty }$\end{document} bound when n ≤ 5 under the very weak assumption that, for some ε > 0, lim inft→∞(tf(t))2-εf'(t)>0\documentclass[12pt]{minimal}
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\begin{document}$\liminf _{t\rightarrow \infty }\frac {(tf(t))^{2-\epsilon }}{f^{\prime }(t)}>0$\end{document} or lim inft→∞t2f(t)f''(t)f'(t)32+ε>0\documentclass[12pt]{minimal}
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\begin{document}$\liminf _{t\rightarrow \infty }\frac {t^{2}f(t)f^{\prime \prime }(t)}{f^{\prime }(t)^{\frac {3}{2}+\epsilon }}>0$\end{document}.
