5 problems

Let $u\colon\mathbb{R}^{n}\to[-1,1]$ be a solution of the Allen–Cahn equation in $\mathbb{R}^{n}$. Call $u$ stable when the second variation of the Allen–Cahn energy is non-negativ…

Let $\Omega\subset\mathbb{R}^N$ be the domain and let $u\in W_0^{1,2}(\Omega)$ be a stable weak solution of the elliptic problem considered in the paper. Assume that $f$ is positiv…

Let $\Omega$ be a domain and consider a solution $u$ of a semilinear Dirichlet problem … with $u>0$ in $\Omega$ and $u=0$ on $\partial\Omega$. A solution is stable in the usual var…

Consider the semilinear elliptic equation … where $f$ is a $C^1$ function, and let $u$ be a bounded stable $C^2$ solution, meaning that … for every…

Conjecture. If $n\leq 9$, then $u$ is bounded.