A priori estimates for semistable solutions of semilinear elliptic equations

TL;DR

This paper establishes new a priori bounds for semistable solutions of semilinear elliptic equations, extending the dimension range up to 5 and 6 under specific growth conditions on the nonlinearity.

Contribution

It proves $L^ty$ bounds for semistable solutions in higher dimensions under growth conditions on $f$, improving previous results limited to lower dimensions.

Findings

Boundedness of solutions up to dimension 5 under growth condition on $f'$.

Extension of boundedness results to dimension 6 with more restrictive assumptions.

No lower bound on $f'$ required for the estimates.

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 $n\leq 9$, but only established for $n\leq 4$. In this paper we prove the $L^\infty$ bound up to dimension $n=5$ under the following further assumption on $f$: for every $\varepsilon>0$, there exist $T=T(\varepsilon)$ and $C=C(\varepsilon)$ such that $f'(t)\leq Cf(t)^{1+\varepsilon}$ for all $t>T$. This bound follows from a $L^p$-estimate for $f'(u)$ for every $p<3$ and $n\geq 2$. 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 $f'$.