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

TL;DR

This paper demonstrates that for stable solutions of semilinear elliptic equations, interior $L^q$ bounds cannot be established in any dimension when relating the $L^p$ and $L^q$ norms of solutions, generalizing previous results that required nonnegativity of $f$.

Contribution

It proves the impossibility of interior $L^q$ estimates for stable solutions with general nonlinearities, extending known results beyond the nonnegative case.

Findings

Interior $L^q$ bounds do not hold for general nonlinearities.

Previous estimates relied on nonnegativity of $f$, which is not necessary now.

The result applies in all dimensions $N \\geq 1$.

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}, Cabr\'e, 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 H\"older regularity for dimensions $N\leq 9$. 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 $N\geq 1$, to obtain an interior $L^q$ estimate in terms of the $L^p$-norm of $u$ whenever $1\leq p<q\leq \infty$.