Consider the steady-state of the reaction-diffusion equation with an inhomogeneous Neumann boundary condition:

$$ \begin{cases} D\Delta u +R(u,x)=0 & \text{in } U, \\ \partial_{\nu}u=f & \text{on } \partial U \\ u \geq 0 & \text{in } U \tag{1} \label{1}, \end{cases} $$

where $U \subset \mathbb{R}^3$ is a smooth bounded open set, and $\nu$ is the outward unit normal.

Here $u = u(x)$ is the concentration of the substance, $D$ is the diffusion coefficient, $R(u,x)$ is the (nonlinear) reaction term, representing local production or decay, and $f$ is some given function.

I could not find any reference that establishes the necessary conditions on $R$ and $f$ for the existence and regularity of solutions to \eqref{1}. Standard books cover linear Elliptic PDEs in detail, but I could not find any literature that treats a general semilinear problem with Neumann Boundary conditions.

Could anyone suggest a book/paper that addresses this? Thanks!

Consider the steady-state of the reaction-diffusion equation with an inhomogeneous Neumann boundary condition:

$$ \begin{cases} D\Delta u +R(u,x)=0 & \text{in } U, \\ \partial_{\nu}u=f & \text{on } \partial U \\ u \geq 0 & \text{in } U \tag{1} \label{1}, \end{cases} $$

where $U \subset \mathbb{R}^n (n \geq 3)$ is a smooth bounded open set, and $\nu$ is the outward unit normal.

Here $u = u(x)$ is the concentration of the substance, $D$ is the diffusion coefficient, $R(u,x)$ is the (nonlinear) reaction term, representing local production or decay, and $f$ is some given function.

I could not find any reference that establishes the necessary conditions on $R$ and $f$ for the existence and regularity of solutions to \eqref{1}. Standard books cover linear Elliptic PDEs in detail, but I could not find any literature that treats a general semilinear problem with Neumann Boundary conditions.

Could anyone suggest a book/paper that addresses this? Thanks!