Suppose $U$ is a bounded domain in $\mathbb{R}^n$ ($n \geq 3$) with smooth boundary. Consider the semilinear Elliptic PDE:
$$ \begin{cases} -\Delta u=f(x,u) & \text{in } U, \\ \partial_{\nu}u=g & \text{on } \partial U \tag{1} \label{1}, \end{cases} $$ where $f: U \times \mathbb{R} \to \mathbb{R}$ is nonlinear, and $g: \partial U \to \mathbb{R}$ are given functions, and $\nu$ is the unit outer normal at each point on $\partial U$.
Q1. Under what conditions on $f$ and $g$, does a weak solution to \eqref{1} exist, i.e. $\exists u \in H^1(U)$ such that
$$\int_{U} \nabla u \cdot \nabla v \,dx=\int_U fv \, dx + \int_{\partial U} gv \, ds \quad \forall v \in H^1(U).$$
Clearly, $f$ and $g$ must satisfy the compatibility condition:
$$\int_U f \, dx =- \int_{\partial U} g \, ds $$
But is this sufficient?
Q2. Under what assumptions on $f$ and $g$, can we show that (i) the weak solution $u$ is in $ H^2(U)$, (ii) recover a classical solution?
Remark: In the discussion here, it is shown that $u \in W^{2,p}(U)\cap C^{1,\alpha}(U)$ when $|f(x,u)| \leq C(1+|u|^p)$.
Q3. Has \eqref{1} been studied for general $f$ with appropriate growth/regularity constraints? Any reference that covers the existence and regularity of its solutions?
