I would like to know whether it is possible to make the following construction: Construct two functions $$\sigma \in C^1(\mathbb{R}^n \setminus \{0\}), \quad \sigma(x) > 0, \quad \sigma(x) \to 0 \text{ as } |x| \to 0, \text{ and} $$ $$ \quad \text{continuous } \varphi : (0,\infty) \to (0,\infty), \quad \int^{\varepsilon}_0 \frac{1}{\varphi(t)}dt = \infty \text{ for all $\varepsilon > 0$,} $$
such that
$$|\nabla \sigma(x) |^2 \le \varphi^2(\sigma(x)), \qquad \text{all $x \in \mathbb{R}^n$.} $$
One can get pretty close to this construction by choosing $\sigma(x) = e^{-1/|x|}$ and setting $\varphi(x)$ (near zero) to be $x|\log(x)|$, but this is not good enough. The condition that the integral of $1/\varphi$ must diverge on any neighborhood of zero seems to be most restrictive of the above conditions.
I am interested in such a construction because, it turns out that, the existence of such a pair $(\sigma, \varphi)$ can help one show that certain Schrödinger operators $-\Delta + V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, with potentials $V$ that are singular at zero, are essentially self adjoint on $C_0^\infty(\mathbb{R}^n \setminus\{0\})$. See Stetkær-Hansen 1966, Mathematica Scandinavica.