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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.

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  • $\begingroup$ I would search "semilinear elliptic equation." There are many classical results due to Gidas, Nirenberg, Spruck, Wei-Ming Ni, etc. $\endgroup$
    – sharpend
    Commented Aug 9, 2020 at 5:29

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The case $p=2$ is the nonlinear Schrödinger 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.

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  • $\begingroup$ is $u$ here a scalar or vector? $\endgroup$
    – Math604
    Commented Aug 9, 2020 at 17:44
  • $\begingroup$ both cases are considered in the literature, for the vectorial nonlinear Schrödinger equation, see for example arxiv.org/abs/1110.2990 $\endgroup$
    – Carlo Beenakker
    Commented Aug 9, 2020 at 18:31
  • $\begingroup$ do you have any references for the purely elliptic problem on a bounded domain with $u=0$ on $ \partial \Omega$. (ie. $-\Delta u= |u|^p u$ in $\Omega$) where $u$ a vector and where they discuss existence of nonzero solutions and $p$. $\endgroup$
    – Math604
    Commented Aug 9, 2020 at 18:41

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