Given a connected bounded open domain $U$ with a smooth boundary and the following homogeneous Neumann boundary-value problem \begin{cases} \nabla [a(x) \nabla u(x)] = 0 & \text{in } U \newline \frac{\partial u}{\partial \nu} = 0 & \text{on } \partial U \end{cases} where $a:U\mapsto\mathbb{R}^+$ is a smooth function.
Are the only smooth solutions constant?
I have found several proofs and statements for constant $a(x)$, i.e., for the Laplace equation, but none for elliptic PDEs. It sounds like a basic problem to me, but I did not find it in the book by Gilbarg and Trudinger and neither in that by Evans. Of course, I also searched in math stackexchange and beyond, but without success. Since I am new to the study of PDEs, I may miss essential keywords or subtle details.
A reference would be a great help to me.
If $a \in L^\infty(U)$ then $u$ is a constant. Indeed, if you multiply your equation by the solution $u$ then integrate by parts then you obtain that $$0=\int_U u \operatorname{div}(a\nabla u ) \, dx =-\int_Ua \vert \nabla u \vert^2\,dx + \int_{\partial U}au \frac{\partial u}{\partial \nu} \, dx =-\int_Ua \vert \nabla u \vert^2\,dx.$$ Since $a>0$ and $U$ is connected, we must have that $u$ is a constant in $U$.