Let $\Omega$ be a bounded domain with smooth boundary . Let $u$ be a solution of the problem $\Delta u+ a_i(x)\frac{\partial u}{\partial x_i}=f(x)$ and $\frac{\partial u}{\partial n}=0$ . Assume that $f\geq 0$ in $\Omega$ then to show that $u$ is a constant and $f=0$.
I was trying to use maximum principle to get the contradiction for $f>0$ by multiplying with $u$ on both side and using integration by parts but unable to prove as $\int_{\Omega} a_i(x) u\frac{\partial u}{\partial x_i}$ depends on sign of $a_i(x)$ Any hints or suggestion would be helpful. It is a problem of Renardy Rogers chapter 4.
Let $u \in C^2(\Omega) \cap C^1(\overline \Omega)$. Since $f\geqslant 0$, $$ \Delta u +\sum_{i=1}^na_i(x)\partial_iu(x)\geqslant 0\qquad \text{in } \Omega.$$
Let $M= \max_{\overline \Omega} u$ and consider the set $\{u=M\}:=\{x\in \overline \Omega \text{ s.t. } u=M\}$. By the extreme value theorem, $\{u=M\} $ is non-empty. We claim that $$\{u=M\} \cap \Omega \neq \varnothing. \tag{$\ast$}$$ Suppose that, for the sake of contradiction, this is not the case. Then $\{u=M\} \subset \partial \Omega$ and so $u(x)<\max_{\overline \Omega} u$ for all $x\in \Omega$ and there exists $x_0\in \partial \Omega$ such that $u(x_0)=\max_{\overline \Omega} u$. But then Hopf's lemma, implies that $\partial_\nu u(x_0)<0$ which contradictions the Neumann boundary condition. Thus, we have proven $(\ast)$.
But this implies that there exists $x_0\in \Omega$ such that $u(x_0)=\max_{\overline \Omega} u$ so, by the strong maximum principle, $u$ is a constant.
Finally, since $u$ is a constant, $$f = \Delta u +\sum_{i=1}^na_i(x)\partial_iu(x) =0.$$