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I'm looking for the solution of the following equation

$$2cE^3=E_u^2+ E_v^2$$ where $c$ is a constant and $E$ is an harmonic function w.r. to the variables $u$ and $v$.

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I assume that you are taking $E$ to be a harmonic function of $(u,v)$. If $c$ is not zero, then the only solution is $E=0$. If $c=0$, then the only solutions are to have $E$ be constant. One proves this by differentiating the equation a couple of times and solving for the higher partials in terms of $E$ and its first partials.

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