A priori estimates for semistable solutions of semilinear elliptic equations

Abstract

We consider positive semistable solutions $u$ of $Lu+f(u)=0$ with zero Dirichlet boundary condition, where $L$ is a uniformly elliptic operator and $f\in C^2$ is a positive, nondecreasing, and convex nonlinearity which is superlinear at infinity. Under these assumptions, the boundedness of all semistable solutions is expected up to dimension $n\le 9$, but only established for $n\le 4$. In this paper we prove the $L^{\infty }$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\epsilon >0$, there exist $T=T(\epsilon )$ and $C=C(\epsilon )$ such that $f^{\prime }(t)\le Cf(t{)}^{1+\epsilon }$ for all $t>T$. This bound follows from a $L^p$-estimate for $f^{\prime }(u)$ for every $p<3$ and $n\ge 2$. Under a similar but more restrictive assumption on $f$, we also prove the $L^{\infty }$ estimate when $n=6$. We remark that our results do not assume any lower bound on $f^{\prime }$.

One bibliographical reference has been corrected with respect to the previous version

Cited by in corpus (1)

• Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities