New a priori estimates for semistable solutions of semilinear elliptic equations

TL;DR

This paper derives new a priori estimates for semistable solutions of semilinear elliptic equations, leading to bounds on solutions in certain dimensions under specific growth conditions on the nonlinearity.

Contribution

It introduces novel integral estimates for semistable solutions, enabling improved a priori bounds in dimensions up to 9 and 5 under new growth assumptions.

Findings

Established integral estimates for semistable solutions.

Derived $L^{ ext{infinity}}$ bounds in dimensions $n \\leq 9$ and $n \\leq 5.

Provided conditions on $f$ ensuring bounded solutions.

Abstract

We consider the semilinear elliptic equation $-L u = f(u)$ 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 $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. We prove that if $u$ is a positive semistable solution then for every $0\leq\beta<1$ 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 $n\leq 9$, under the extra assumption that $\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}} < \frac{2}{9-2\sqrt{14}}\cong 1.318$. Also, we establish a priori $L^{\infty}$ bound when $n\leq 5$ under the very weak assumption that, for some $\epsilon>0$, $\liminf_{t\rightarrow\infty} \frac{(tf(t))^{2-\epsilon}}{f'(t)} > 0$ or $\liminf_{t\rightarrow\infty} \frac{t^{2}f(t)f"(t)}{f'(t)^{\frac{3}{2}+\epsilon}} > 0$.