I have to check that a specific functions solves a PDE and I can't find the mistake in my calculations, so if anyone could help me out with this I would be very thankful! So the problem is as follows:
Given two smooth harmonic conjugate functions $u$ and $v$ in $\mathbb{D}$ and $a>0$ we have to show that the function $$\varphi= \frac{1}{a}\log\left(\frac{2(u_x^2+u_y^2)}{a(1+u^2+v^2)^2}\right)$$ solves the nonlinear elliptic equation $-\Delta\varphi = e^{a\varphi}$ in $\mathbb{D}$, where $\mathbb{D}$ is the open unit disc and $\Delta = \partial_x^2 + \partial_y^2$ means the Laplace operator.
I interpreted harmonic conjugate functions to be two functions that solve the Cauchy-Riemann equations, i.e. $u_x = v_y$ and $u_y = -v_x$ but we never really defined them in our lectures.
So my way of solving is the following: I define $f:=u_x^2+u_y^2$ and $g:=1+u^2+v^2$ to simplify the function as $$\varphi = \frac{1}{a}\log\left(\frac{2f}{ag^2}\right).$$ From this I get $$\Delta\varphi = \frac{1}{a}\left(\frac{f\cdot\Delta f - f_x^2-f_y^2}{f^2}-2\frac{g\cdot\Delta g - g_x^2-g_y^2}{g^2}\right) \tag{1}\label{1}$$ and $$e^{a\varphi} = \frac{2f}{ag^2} \tag{2}\label{2}$$ and now I can do the tedious derivatives in parts. For the first term I get $$\Delta f = 4(u_{xy}^2+v_{xy}^2)$$ where I used that $\frac{\partial}{\partial x}\Delta u = \frac{\partial}{\partial y}\Delta u = 0$ and that $u_{xx}^2 = u_{yy}^2 = v_{xy}^2$, which we know from both being harmonic conjugates. This leads to $$f\cdot\Delta f - f_x^2-f_y^2 = -8u_x u_y u_{xy}\cdot\Delta u = 0.$$ With this and equations \eqref{1} and \eqref{2} I get that for $\varphi$ to be a solution the following must hold: $$g\cdot\Delta g - g_x^2-g_y^2 = f.$$ Now again by using some properties of harmonic conjugates, this time just the Cauchy-Riemann equations and that $u$ and $v$ are harmonic, and substituting $f$ back in I get $$\begin{align} \Delta g = 4f, &&g_x^2+g_y^2 = 4f\,(u^2+v^2) \end{align}$$ which leads to $$g\cdot\Delta g - g_x^2-g_y^2 = 4f \neq f$$
I don't know where the mistake is or if I missed something. I also started trying the same method in polar coordinates with the polar version of laplace operator but that seemed to get me to the same result. Maybe it helps to know that this is a course on PDEs and maximum principles and symmetry.