2D semilinear elliptic PDE

Question

This is the simplest equation arising from a fascinating (to me) and obscure vector field theory of mathematical physics first developed in 1962, and for which no solutions have ever been found. The following PDE is the corrected version (June 13, 2017) from the original post (February 25, 2015), which had a sign error. This PDE is the result of 47 years of work for me, so any help would be appreciated since I am not a professional mathematician or physicist. $$u_{rr}+u_{zz}+u_{r}/r-u/r^2+u(u_r ^2+u_z ^2+uu_r/r)=0 $$where (r,z) are cylindrical coordinates and u(r,z) is a dimensionless azimuthal velocity field defined for all (Euclidean) space, with conditions \begin{cases}0\leq r\lt \infty\\-\infty\lt z\lt\infty\\0\leq u(r,z)\lt\infty\\u(r,z)=u(r,-z)\\ \lim_{r\to0}u(r,z)=0\\ \lim_{r^2+z^2\to\infty}u(r,z)=0\\\end{cases}Lie symmetry analysis yields no valuable results. I have unsuccessfully tried to find an ansatz that would simplify the PDE, and possibly lead to a closed form solution. Numerical analysis cannot be used because the domain is unbounded. I do not think there is a Lagrangian for this PDE. So my question is this: Do solutions exist for this PDE and conditions? And if so, how does one go about finding them? I have an undergraduate degree in physics, so my skills with PDEs are modest. Most texts on elliptic PDEs are beyond my abilities, so any help would be greatly appreciated!

Answer 1

Multiply your equation by $u$ and integrate over $\mathbb{R}^3$ in azimuthal coordinates (so the volume form is $r~\mathrm{d} r~\mathrm{d}z ~\mathrm{d}\theta$ you get

$$ 0 = 2\pi \iint_{r,z} (u_{rr} + u_{zz} + u_r/r) ur - u^3 u_r - r u^2(u_r^2 + u_z^2 + 1/r^2) \mathrm{d}r \mathrm{d}z $$

Integrate by parts the first term (assuming that it is integrable) using the boundary conditions you get

$$ 0 = \iint_{r,z} - (u_r^2 + u_z^2) r - \frac14 \partial_r(u^4) - r u^2(u_r^2 + u_z^2 + 1/r^2) \mathrm{d}r \mathrm{d}z $$

The middle term integrates to zero since $u^4 = 0$ when $r = 0$ and $r \to \infty$. The first and last terms are manifestly negative. Hence the only solution is $u \equiv 0$.

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Equivalently, extend your function to a rotationally symmetric function on $\mathbb{R}^3$: that is $u = u(r,z,\theta) = u(r,z)$.

Your equation can be re-written as

$$ \triangle u - \frac{u^2}{r} \partial_r u = u(u_r^2 + u_z^2 + 1/r^2) > 0$$

as $u$ is non-negative. And by assumption that $u = 0$ when $r = 0$ indicates that under a regularity assumption $u^2 / r$ is bounded above and below. So we can apply the maximum principle, which with the boundary condition that $u\to 0$ as $r^2 + z^2 \to \infty$ shows that $u$ must vanish identically.

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If the sign of the last term in your equation is $+$ and not $-$, then neither of the above argument would work, and then the equation can potentially have a solution (requires more analysis). (Basically, as it stands you have a repulsive self-interaction term which prevents a bound state from existing. Changing the sign makes the self-interaction attractive which then may allow for a bound state.)