Consider $U\subset \Bbb R^n$ an limited open and $u\in H^1(U)$. How i can define the weak soluton of the problem $$\Delta u=0.$$
$(1)$ $$\int_U \nabla u\nabla \phi=0,$$ for every $\phi\in C^{\infty}_c(U)$. But this implies that since $Du\in L^2(\Omega)$, there exists a sequence $\phi_n\in C_c^{\infty}(\Omega)$ such that $\nabla \phi_n\to \nabla u$ in $L^2(\Omega)$. Then, $$\int_{\Omega}|\nabla u|^2=\lim_{n\to\infty}\int_{\Omega}\nabla u\nabla \phi_n=0,$$ so $u$ is constant in every connected component of $\Omega$(?). is it possible that i have only costant solutions? Where is my error?
You must specify the boundary conditions for your PDE problem in order to define a weak solution.
Your definition of weak solution is for the PDE problem $\Delta u = 0 \text{ on } U$, $u=0 \text{ on } \partial U$. The only solution to this PDE problem is the zero solution, which can be easily seen (for example) by the maximum principle.