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Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy $$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$ where $f$ is a fast decaying smooth function.

I would like to know how the solutions depend on $\alpha$. Is $u$ a continuous or differentiable function with respect to $\alpha$? I will also appreciate any reference.

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  • $\begingroup$ Are you able to understand the one-dimensional case or the radial case in $\mathbb{R}^2$? $\endgroup$
    – Dispersion
    Commented Mar 19, 2023 at 16:27
  • $\begingroup$ I am not! However, I would appreciate any insight even for the radial case. $\endgroup$
    – A random mathematician
    Commented Mar 19, 2023 at 16:38
  • $\begingroup$ What do you know about the well-posedness theory for these equations? $\endgroup$
    – cs89
    Commented Mar 19, 2023 at 19:17

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