Weak maximum principle for $W^{1,p}$ weak solution to elliptic equations

Question

Let $\Omega\subset \mathbb R^n$ be a bounded open subset and $a^{ij}\in L^\infty(\Omega)$ satisfy $a^{ij}\xi_i\xi_j\ge \lambda |\xi|^2$ and $\|a^{ij}\|_{L^\infty}\le \Lambda$. Let $1<p<2$ and $u\in W_0^{1,p}(\Omega)$ solves $D_j(a^{ij}D_i u)=0$ weakly in the sense that $$\int_\Omega a^{ij}D_iuD_j\phi dx=0\quad \forall \phi\in C_c^{\infty}(\Omega).$$ I wonder if $u$ must be $0$ in this case. This is true if we assume $u\in W_0^{1,2}(\Omega)$ by weak maximum principle of weak solution cf. Gilbarg-Trudinger Theorem 8.1. However, the test function it takes does not work in the $W_0^{1,p}(\Omega)$ setting since the integral does not make sense.