This is not a technical mathematical question. I came across some PDEs with no references nor their names.

$$-\Delta u + \int_\Omega udx = f\qquad \hbox{in $\Omega$} \label{1}\tag{Eq1}$$

The above equation can be augmented either with Dirichlet boundary condition $u=g$ on $\partial\Omega$ or with Neumann boundary condition $\partial_n u=g$ on $\partial\Omega$.

I would like to know the name and the application of this type of problem. Or any good reference to this.

A slightly more general setting would be considering $$-\Delta u + \varphi\Big(\int_\Omega udx\Big) = f\qquad \hbox{in $\Omega$} \label{2}\tag{Eq2}$$ For an appropriate function $\varphi$. I have the feeling this must have a good application that because assuming, for instance, $\varphi=0$ and \eqref{2} is augmented with the Neumann boundary condition $\partial_n u=g$ on $\partial\Omega$ then it becomes the classical Neumann problem whose solutions exist if and only if we have the compatibility condition

$$ \int_\Omega fdx+ \int_{\partial\Omega} gdx =0.\label{c}\tag{C}$$

In practice, $g$ is the flux term, $f$ is the source term. **I forgot the meaning of the compatibility condition \eqref{c}.**

**Question 1:** What is the name and application of the problem \eqref{1} or \eqref{2}. Or what are references to this problem?

Another problem is the following Dirichlet problem with $f\in L^p(\Omega)$ $1<p<\infty$

$$-\Delta u + u = f\qquad \hbox{in $\Omega$} \label{3}\tag{Eq3}$$

Brezis' book claims there is a unique $u\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$ solving \eqref{3} and we have $\|u\|_{W^{2,p}(\Omega}\leq c\|f\|_{L^p(\Omega}$

**Question 2:** what are the references to this type of problem?

rank-one perturbation of the Poisson equation.$\endgroup$