Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian

Abstract

We consider stable solutions to the equation $ -\Delta_p u =f(u) $ in a smooth bounded domain $\Omega\subset\mathbb{R}^n $ for a $ C^1 $ nonlinearity $f$. Either in the radial case, or for some model nonlinearities $f$ in a general domain, stable solutions are known to be bounded in the optimal dimension range $n<p+4p/(p-1)$. In this article, under a new condition on $n$ and $p$, we establish an $ L^\infty $ a priori estimate for stable solutions which holds for every $ f\in C^1$. Our condition is optimal in the radial case for $n\geq3$, whereas it is more restrictive in the nonradial case. This work improves the known results in the topic and gives a unified proof for the radial and the nonradial cases. The existence of an $L^\infty$ bound for stable solutions holding for all $C^1$ nonlinearities when $n<p+4p/(p-1)$ has been an open problem over the last twenty years. A forthcoming paper by Cabr\'e, Sanch\'on, and the author will solve it when $p>2$.