New a priori estimates for semistable solutions of semilinear elliptic equations

Abstract

We consider the semilinear elliptic equation $-Lu=f(u)$ in a general smooth bounded domain $\Omega \subset R^n$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f$ is a $C^2$ positive, nondecreasing and convex function in $[0,\infty )$ such that $\frac{f(t)}{t}\to \infty$ as $t\to \infty$. We prove that if $u$ is a positive semistable solution then for every $0\le \beta <1$ we have $$f(u){\int }_0^{u}f(t)f"(t)e^{2\beta {\int }_0^t\sqrt{\frac{f"(s)}{f(s)}}ds}dt\in L^1(\Omega ),$$ 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 $L^{\infty }$ bound in dimensions $n\le 9$, under the extra assumption that ${\mathrm{limsup}}_{t\to \infty }\frac{f(t)f"(t)}{f^{\prime }(t{)}^2}<\frac{2}{9-2\sqrt{14}}\cong 1.318$. Also, we establish a priori $L^{\infty }$ bound when $n\le 5$ under the very weak assumption that, for some $\epsilon >0$, ${\mathrm{liminf}}_{t\to \infty }\frac{(tf(t){)}^{2-\epsilon }}{f^{\prime }(t)}>0$ or ${\mathrm{liminf}}_{t\to \infty }\frac{t^{2}f(t)f"(t)}{f^{\prime }(t{)}^{\frac{3}{2}+\epsilon }}>0$.

arXiv comment15 pages