A new proof of the boundedness results for stable solutions to semilinear elliptic equations

We consider the class of stable solutions to semilinear equations $-\Delta u=f(u)$inaboundedsmoothdomainof$$\mathbb{R}$^n$. Since 2010 an interior a priori $L^{\mathrm{\infty }}$ bound for stable solutions is known to hold in dimensions $n \leq 4$forall$C^1$nonlinearities$f$. In the radial case, the same is true for $n\le 9$. Here we provide with a new, simpler, and unified proof of these results. It establishes, in addition, some new estimates in higher dimensions —for instance $L^p$ bounds for every finite~$p$ in dimension 5. Since the mid nineties, the existence of an $L^{\mathrm{\infty }}$ bound holding for all $C^1$ nonlinearities when $5\le n\le 9$ was a challenging open problem. This has been recently solved by A. Figalli, X. Ros-Oton, J. Serra, and the author, for nonnegative nonlinearities, in a forthcoming paper.