# Klein Gordon equation - references

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?

• Klein-Gordon ? Did you google that name ? Nov 3, 2018 at 10:01
• By the way, I did my PhD thesis on this equation (plus a source term). That was in 1978. Nov 3, 2018 at 10:02
• 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 $p = 2$ Nov 3, 2018 at 11:05
• 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. Nov 3, 2018 at 11:54
• 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... Nov 3, 2018 at 14:33

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},
}

• Is there a version time-independent in which$\Delta u + \lambda u^p=0$ with $u$ a scalar function of two spatial variables? Nov 30, 2018 at 20:27

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.

• 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 Nov 3, 2018 at 15:54
• @Denis Serre - in your PhD thesis, is the pde not time-dependent? Nov 30, 2018 at 20:53
• @AlexanderPigazzini. No, it was not. Dec 1, 2018 at 7:28
• @Denis Serre - Thanks, has a name that kind of pde? Dec 1, 2018 at 13:05