Lack of interior $L^q$ bounds for stable solutions to elliptic equations

Abstract

We consider stable solutions of semilinear elliptic equations of the form $-\Delta u=f(u)$ in a bounded domain $\Omega \subset {\mathbb{R}}^N$. In a well-known paper \cite{cfrs}, Cabré, Figalli, Ros-Oton and Serra obtained interior estimates for the $W^{1,2}$-norm of $u$ in terms of the $L^1$-norm of $u$ and proved interior Hölder regularity for dimensions $N\le 9$. All these results rely on the assumption that $f$ is nonnegative. We show that, for general nonlinearities $f\in C^{\infty }(\mathbb{R})$, it is impossible, in any dimension $N\ge 1$, to obtain an interior $L^q$ estimate in terms of the $L^p$-norm of $u$ whenever $1\le p<q\le \infty$.

9 pages