Solution to Seiberg-Witten monopole equation To understand some physics problem, I want to know if there is (non-$L^2$ or $L^2$) solution to the Seiberg-Witten equation on $\mathbb{R}^4$
\begin{equation}
    {D}_A \psi = 0\\
    F_A^+ = i\zeta\omega + \sigma(\psi)
\end{equation}
where $\zeta$ is a real number, and $\omega$ is the standard symplectic form on $\mathbb{R}^4$.
I know that when $\zeta = 0$, a square-integrable solution would imply $\psi = 0$ everywhere, because by Weitzenbock formula
\begin{equation}
    D_A^*{D_A}\psi = \nabla _A^*{\nabla _A}\psi  + \frac{s}{4}\psi  + \frac{1}{2}F_A^ + \psi 
\end{equation}
and then using the S-W equation itself, one gets
\begin{equation}
    \int {{{\left| {{\nabla _A}\psi } \right|}^2} + \frac{1}{4}{{\left| \psi  \right|}^4} + \left( {i\zeta\omega  \cdot \psi ,\psi } \right)}  =  - \frac{1}{4}\int {s{{\left| \psi  \right|}^2}} 
\end{equation}
which forces $\psi = 0$ when $s = \zeta = 0$

But I wonder what happens if $\zeta \ne 0$? 
Is there known solution (non-$L^2$ is also good) to the equation? Any
  reference would be great.

 A: The noncompactness is bound to create trouble. For what to expect  in the case of a compact symplectic manifold  check  page 272-275 of  these notes  Kronheimer and Mrowka have investigated the noncompact  case  in a beautiful paper going back to the mid 90s.    Their main result seems to say that the answer to your question is positive. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$
Comment 1.  If we think  $\bR^4=\bC^2$, then we have
$$\Psi=\alpha+\beta,\;\;\alpha\in \Omega^0(\bC^2),\;\; \beta\in \Omega^{0,2}(\bC^2). $$
Closing our eyes and pretending  that $\bR^4$  is compact    we deduce  from the  SW equations $\newcommand{\bpar}{\bar{\partial}}$
$$ F_A^{0,2}=0,\;\;\bpar_A \alpha=0=\bpar_A^*\beta, $$
$$\alpha\beta=0, $$
$$ F_A\wedge \omega= \sqrt{-1}(|\alpha|^2-|\beta|^2-\zeta) dV. $$
One of $\alpha$ or $\beta$ is identically zero.  If $F_A\wedge \omega \to 0$ at $\infty$ then   either $|\alpha|^2\to \zeta$ or $|\beta|^2\to-\zeta$ at $\infty$ vilating the $L^2$-assumption on $\Psi$.   As in the paper  of Kronheimer and Mrowka, you need to impose a different condition at $\infty$ to get nontrivial solutions.
