Consider $\Omega \subset \mathbb{R}^{N}$ a smooth domain and $$Lu = -\text{div}(A(x)\nabla u) + b(x) \cdot \nabla u + c(x)u,$$ a uniformly elliptic operator with $A(x) = (a_{ij}), a_{ij} \in C^{1}(\Omega), b = (b_{1}, ..., b_{N}), b_{i} \in L^{\infty}(\Omega)$ and $c \in L^{\infty}(\Omega)$.
My professor gave the following definition for subsolution:
Definition: We say that a function $u \in W^{2,p}(\Omega)$, $p > N$ is a subsolution of $L$ if $$ (1)\,\,\, \begin{cases} L u \leq u, \Omega, \text{a.e.} \\ \,\,\,\,u \leq 0,\Omega. \end{cases} $$
Is it right to think that the natural way to think that $u \in W^{2,p}(\Omega)$ is a subsolution of $L$ is to say that $$ (2)\,\,\, \int_{\Omega} (A \nabla u)\cdot \nabla \phi + (b\cdot \nabla u) \phi + (cu)\phi \leq \int_{\Omega} f(x,u) \phi, \quad \forall \phi \in C^{1}_{0}(\Omega) $$ and $\gamma_{0}(u) \leq 0$ (where $\gamma_0$ is the trace operator)? If I'm right, anyone know any reference with an appropriate version of divergence theorem which I could use to get from $(2)$ to $(1)$ ?