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I am interested in the following: suppose one has proved theoretically the existence of solution $u = u (x_1,...,x_n)$ of some pde. I wonder how can be shown that the obtained solution $u$ is not trivial, meaning that is not constant or depends on all variables $x_1,...,x_n$. Which are the main techniques to prove that kind of things?

Thanks in advance, any answer is welcome.

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