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The Klein Gordon equation of the form: $\Delta u+ \lambda u^p=0$ is been studied for $p = 2$? (i.e.$\Delta u+ \lambda u^2=0$) If yes are there references?

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    $\begingroup$ Klein-Gordon ? Did you google that name ? $\endgroup$ Nov 3, 2018 at 10:01
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    $\begingroup$ By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978. $\endgroup$ Nov 3, 2018 at 10:02
  • $\begingroup$ Thank you for answer, I found this equation (in this form) in a list of nonlinear pde on wikipedia en.m.wikipedia.org/wiki/… and I asked myself if it had been studied for $p = 2$ $\endgroup$
    – exxxit8
    Nov 3, 2018 at 11:05
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    $\begingroup$ By the way, this is probably not what you are asking, but in $5+1$ dimensions, this is the classical equation of the so-called $\phi^3$ theory. The resulting quantum field theory is (was?) a popular toy model because it displays asymptotic freedom, just like (nonabelian) Yang-Mills in $3+1$, which is harder to study. $\endgroup$ Nov 3, 2018 at 11:54
  • $\begingroup$ thats a bit suprising that wikipedia says that...as mentioned above, the name is completely off... Klein Gordon is related to a wave equation..maybe try Lane-Emden (or Lane-Emden-Fowler) (actually i not sure of exact name either) or even try 'semilinear elliptic' and you will get a million hits... $\endgroup$
    – Math604
    Nov 3, 2018 at 14:33

2 Answers 2

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This is a nonlinear Klein-Gordon equation of the type $$ \Delta u = f(u) $$ which, at least in $1+1$ dimensions has been studied for several functions $f$; although the most interesting seem to be when $f$ are exponential or trigonometric, e.g., sine-Gordon equation or approximations thereof.

The case where $f(u) = u^n$ appears in §7.1.1.1 of the Handbook of nonlinear PDEs by Polyanin and Zaitsev.

@book {MR2865542,
    AUTHOR = {Polyanin, Andrei D. and Zaitsev, Valentin F.},
     TITLE = {Handbook of nonlinear partial differential equations},
   EDITION = {Second},
 PUBLISHER = {CRC Press, Boca Raton, FL},
      YEAR = {2012},
     PAGES = {xxxvi+876},
      ISBN = {978-1-4200-8723-9},
   MRCLASS = {35-00 (35C05)},
  MRNUMBER = {2865542},
}
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  • $\begingroup$ Thank you very much, your answer was very helpful $\endgroup$
    – exxxit8
    Nov 3, 2018 at 12:09
  • $\begingroup$ Is there a version time-independent in which$\Delta u + \lambda u^p=0$ with $u$ a scalar function of two spatial variables? $\endgroup$ Nov 30, 2018 at 20:27
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Perhaps I should mention what I did in my PhD thesis. I studied the non-homogeneous case, that is $\Delta u=\lambda u^2+f$, where $f=f(x)$ is the data. It is particularly interesting to consider the dependence of the solution in terms of the parameter $\lambda$. I worked in a bounded domain $\Omega$, with Dirichlet boundary condition $u=0$ on $\partial\Omega$.

Let me call a solution stable if the linear operator $-\Delta+2u$ is positive, that is if $w\mapsto\int_\Omega(|\nabla w|^2+2uw^2)dx$ is equivalently to the (square) norm of $H^1_0(\Omega)$.

Theorem : There exist two numbers $-\infty\le\lambda_-(f)<\lambda_+(f)\le+\infty$ such that the boundary value problem admits a stable solution if and only if $\lambda\in(\lambda_-(f),\lambda_+(f))$. Moreover, this stable solution is unique (denote it $u_\lambda$). For such parameters, there exists at least one non-stable solution. The map $\lambda\mapsto u_\lambda$ is increasing.

It happens that $\lambda_+(f)=+\infty$ iff the solution of the linear equation $\Delta U=f$ is non-positive. At a finite extremity of the interval, the solution still exists (it is the limit of $u_\lambda$) but is marginally stable.

Even more interesting is:

Theorem: The function $\lambda\mapsto u(\lambda)$ is analytic and extends to the upper and lower halves of the complex plane. This extension is continuous up to the real line. When $\lambda\in{\mathbb R}\setminus[\lambda_-(f),\lambda_+(f)]$, this extension provides a (non-real) complex solution which has a stability property, and it is unique in this class, up to complex conjugacy.

See: Prolongement analytique et nombre de solutions d'une équation aux dérivées partielles elliptique non linéaire paramétrée, C. R. Acad. Sci. Paris Sér. A (1978) 287, pp 1021-1023.

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  • $\begingroup$ Thank you very much, it's very very interesting, on the web can I find your PhD thesis? I'd like to read it, if possible $\endgroup$
    – exxxit8
    Nov 3, 2018 at 15:54
  • $\begingroup$ @Denis Serre - in your PhD thesis, is the pde not time-dependent? $\endgroup$ Nov 30, 2018 at 20:53
  • $\begingroup$ @AlexanderPigazzini. No, it was not. $\endgroup$ Dec 1, 2018 at 7:28
  • $\begingroup$ @Denis Serre - Thanks, has a name that kind of pde? $\endgroup$ Dec 1, 2018 at 13:05

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