Klein Gordon equation - references

Question

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?

Answer 1

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},
}

Answer 2

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.