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?



Seeking for references on some PDEs
Some time ago I've addressed a similar problem in [this Q&A](https://mathoverflow.net/questions/322568/laplace-equation-with-integral-source-terms/325079?r=SearchResults&s=7%7C4.4618#325079), so I feel I can offer something useful regarding the posed question.



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


The only reference I know that deals extensively with non-local equations i.e. with integrodifferential equations is the monograph of Chia-Ven Pao [2]. Elliptic integrodifferential equations more general than and including \eqref{1} are analyzed in chapter 3, §3.7, pp. 125-133: note that Pao considers the general boundary condition
$$
\mathbf{B} u(x)\equiv\alpha\_0 (x) \frac{\partial u(x)}{\partial n} + \beta\_o(x) u(x)= h(x)\qquad x\in\partial\Omega
$$
and he develops the theory in Ḧolder spaces, i.e. $C^{k+\alpha}\equiv C^{(k,\alpha)}$, $k\in\Bbb N$ and $\alpha\in (0,1)$. To my knowledge, the equation has not a particular name: however, similar equations are widely used in applied mathematics as, for example, ~~it describes the stationary distribution of neutrons inside a fission reactor~~ in the description of nonlocal combustion problems described in references [A1] and [A3] (cited in [2], chapter 3, §3.8, example 4.c, pp. 137-138; see also other references cited at the end of §3.9, p.138). In nuclear reactor theory, an equation similar to \eqref{1} is obtained in [A2], equation 2.16, p. 51, by integrating the stationary linear Boltzmann equation respect to the velocity direction $\mathbf{s}\in\Bbb S^3$: however, the authors do not use the equation in their subsequent development, perhaps because the structure of the integral operator obtained is very complex.



> 
> **Question 2:** what are the references to this type of problem (i.e. \eqref{3})?
> 
> 
> 


Reference [1] chapter IV, §1.2, pp. 172-174, cited in [this Q&A](https://math.stackexchange.com/questions/2940989/need-good-reference-or-a-proof-on-regularity-of-solution-to-neumann-problem/2955480#2955480), gives a proof of the result cited by Brezis for $p=2$ for the more general operator
$$
\nabla\cdot\big(k(x) \nabla u(x)\big) +a(x) u(x)= f(x).
$$
For $p\neq 2$, perhaps it is possible to find the result in the classical monograph of Pierre Grisvard, but I am not sure and I should check.


**References**


[1] V. P. Mikhailov (1978), *[Partial differential equations](https://archive.org/details/MikhailovPartialDifferentialEquations)*, Translated from the Russian by P.C. Sinha. Revised from the 1976 Russian ed., Moscow: Mir Publishers, pp. 396, [MR0601389](http://www.ams.org/mathscinet-getitem?mr=MR0601389), [Zbl 0388.3500](https://zbmath.org/?q=an%3A0388.35002). [Link on Archive.org](https://archive.org/details/MikhailovPartialDifferentialEquations).


[2] Chia-Ven Pao (1992), *Nonlinear Parabolic and Elliptic Equations*, Plenum Press, xv+777, [MR1212084](http://www.ams.org/mathscinet-getitem?mr=MR1212084), [Zbl 0777.35001](https://zbmath.org/?q=an%3A0777.35001). [Free Springer Link](https://link.springer.com/book/10.1007/978-1-4615-3034-3).


**Additional references**


[A1] J. Bebernes, R. Ely, "Comparison techniques and the method of lines for a parabolic functional equation", (English) Rocky Mountains Journal of Mathematics 12, 723-733 (1982), [MR0683864](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0683864), [Zbl 0536.65096](https://www.zbmath.org/?q=an%3A0536.65096).


[A2] Sergio Gallone, Giancarlo Ghilardotti, *Fisica dei reattori nucleari: diffusione e rallentamento dei neutroni* [Nuclear reactor physics: diffusion and slowing down of neutrons], Saggi scientifici 8, Milano: Feltrinelli, 1964, pp. 303.


[A3] C. V. Pao, "Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory", (English), Journal of Mathematical Analysis and Applications 166, No. 2, 591-600 (1992), [MR1160947](https://mathscinet.ams.org/mathscinet-getitem?mr=MR1160947), [Zbl 0762.35049](https://www.zbmath.org/?q=an%3A0762.35049).


