Consider an equation like


$$-\Delta u = |u|^p u $$ in $\Omega$ with $u=0$ on $ \partial \Omega$ where $\Omega$ a domain in $ R^N$ and $ u:\Omega \rightarrow R^N$. Here $p$ is arbitrary or maybe $p=2$. Or consider Neumann problems like this with a zero order term $u$ added to the left.


Do these equations have a name? (I am interested in what kind of results are known and I have no clue what to google). thank you very much.



name of elliptic pde with a power law nonlinearity
The case $p=2$ is the [nonlinear Schrödinger equation,](https://en.wikipedia.org/wiki/Nonlinear_Schr%C3%B6dinger_equation) more generally written as
$$Eu=-\Delta u+\kappa|u|^2 u,$$
with coefficients $E,\kappa\in\mathbb R$. It describes the propagation of light in nonlinear optical fibers and is also a model for a superfluid. In the one-dimensional case the equation can be solved exactly for $\kappa<0$. It supports socalled "soliton" solutions, localized in space.  

 As written the differential equation applies to harmonic solutions $\propto e^{-iEt}$ in the time domain; alternatively, consider functions of both space and time and replace the left-hand-side of the equation by $i\partial u/\partial t$.


Generalizations with $p$ an even integer are also studied in this context, see for example [Schrödinger equation with a power-law nonlinearity.](http://eqworld.ipmnet.ru/en/solutions/npde/npde1402.pdf)


