Consider the simplest elliptic-Neumann problem in $\Omega\subset \mathbb{R}^n$: $$ -\Delta u+u=f\quad \text{in } \Omega, \quad \frac{\partial u}{\partial \nu}=0\quad \text{on } \partial \Omega. \tag{EN} $$ Then the standard $W^{2,p}$-estimate gives $$ f\in L^p(\Omega)\Longrightarrow u\in W^{2,p}(\Omega).\tag{W2p} $$ I blearily remember that $p\geq 1$ in (W2p). Is this indeed required?
Next, for a general elliptic-Neumann problem in $\Omega$: $$ -\Delta u+c(x)u=f\quad \text{in } \Omega, \quad \frac{\partial u}{\partial \nu}=0\quad \text{on } \partial \Omega. $$ What is the (known) mildest (regularity) condition on $c$ so that the $W^{2,p}$-estimate (W2p) holds? In particular, whether $c\in L^q(\Omega)$ (for some $q>0$) would be sufficient to get (W2p)? Any help is highly appreciated!