Give an example of function $u:\mathbb{R}^n\to \mathbb{R}$ such that $\frac{\partial u}{\partial x_i}:\mathbb {R} \to \mathbb {R} \in L^m (\mathbb {R^n}), \ i=1,2, \cdots m$ but $u \notin L^p (\mathbb {R^n}), \ 1 \leq p < \infty$.
My try: take $$u(x)=\int_{-\infty}^{x_1} \phi(s_1)ds_1+\cdots + \int_{-\infty}^{x_n} \phi(s_n)ds_n$$ where $\\$ $$ \phi(x) = \begin{cases} \text{exp} (\frac {1}{x^2-1}) , ~~~~~ x \in (-1,1) \\ 0 , ~~~~~~~~~~~~~~~~~~~~ \text{elsewhere} \end{cases}$$ But I can't show the last condition.