Example of functions in $W^{1,n}(\mathbb{R}^n)$ but not in $L^1(\mathbb{R}^n)$?

Question

Could anyone come up with an example of functions in $W^{1,n}(\mathbb{R}^n)$ but not in $L^1(\mathbb{R}^n)$?

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When $n=1$, there is nothing to say. I don't see an immediate example for $n>2$.

This is motivated by the following remark in Evans's Partial Differential Equations. The notation $W^{1,n}(\mathbb{R}^n)\cap L^1(\mathbb{R}^n)$ suggests that there must some nonintegrable functions in $W^{1,n}(\mathbb{R}^n)$ in general.

Answer 1

Consider the example \begin{align} f(x) = \frac{1}{1+|x|^{3/2}} \end{align} where \begin{align} \int_{\mathbb{R}^n} \frac{dx}{1+|x|^{3/2}} =\int^\infty_0 \int_{|x|=r}\frac{1}{1+r^{3/2}}\ dSdr = C \int^\infty_0 \frac{r^{n-1}}{1+r^{3/2}}\ dr =\infty \end{align} since $n\geq 2$.

Next, observe \begin{align} \nabla f = -\frac{3}{2}\frac{x}{|x|^{1/2}(1+|x|^{3/2})^2} \end{align} which means \begin{align} |\nabla f| = \frac{3}{2} \frac{|x|^{1/2}}{(1+|x|^{3/2})^2} \end{align} which means \begin{align} \int_{\mathbb{R}^n} \frac{dx}{(1+|x|^{3/2})^n} = \int^\infty_0 \int_{|x|=r} \frac{1}{(1+r^{3/2})^n}\ dSdr = C\int^\infty_0\frac{r^{n-1}}{(1+r^{3/2})^n}\ dr <\infty \end{align} and \begin{align} \int_{\mathbb{R}^n} \frac{|x|^{n/2}dx}{(1+|x|^{3/2})^{2n}} = C\int^\infty_0 \frac{r^{3n/2-1}}{(1+r^{3/2})^{2n}}\ dr<\infty \end{align} when $n\geq 2$.

Hence $f \in W^{1,n}(\mathbb{R})$ but $f \not\in L^{1}(\mathbb{R}^n)$.