2 problems

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$, and let $u$ solve … Assume that $u$ has finitely many critical points and that $u(x)>0$ implies $x_1>0$. Half-space rigidity conj…

For the scalar semilinear equation, let $n$ denote the spatial dimension, $m$ the physical mass, and let the nonlinearity exponent satisfy $\alpha\in(0,\infty)$. Forbidden physical…