New a priori estimates for semistable solutions of semilinear elliptic equations

We consider the semilinear elliptic equation $-Lu=f(u)$ in a general smooth bounded domain $\mathrm{\Omega }\subset R^n$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator 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 $t\to \mathrm{\infty }$. We prove that if $u$ is a positive semistable solution then for every $0\le \beta <1$ we have $$f(u){\int }_0^{u}f(t)f\mathrm{"}(t)e^{2\beta {\int }_0^t\sqrt{\frac{f\mathrm{"}(s)}{f(s)}}ds}dt\in L^1(\mathrm{\Omega }),$$ by 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 $L^{\mathrm{\infty }}$ bound in dimensions $n\le 9$, under the extra assumption that $\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f’(t)^{2}} < \frac{2}{9-2\sqrt{14}}\cong 1.318$\mathrm{.}Also,weestablishapriori$L^{\infty}$boundwhen$n\leq 5$undertheveryweakassumptionthat,forsome$\epsilon>0$,$\liminf_{t\rightarrow\infty} \frac{(tf(t))^{2-\epsilon}}{f’(t)}

0$or$\liminf_{t\rightarrow\infty} \frac{t^{2}f(t)f"(t)}{f’(t)^{\frac{3}{2}+\epsilon}} > 0$.