Does there exist a functional energy $I$ such that $$(e^{u}+1)\Delta u+u=0$$ is the Euler-Lagrange equation associated with the energy functional $I$?
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$\begingroup$ Try writing this as $\Delta u+ ue^{-u}=0$ $\endgroup$– AlexArvanitakisCommented Nov 25, 2019 at 0:31
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$\begingroup$ Thank you! If we do not rewritten the equation, is there energy function? $\endgroup$– lidingCommented Nov 25, 2019 at 1:12
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1$\begingroup$ If you don't allow for a rewriting of the equation, then you haven't formulated your question accurately. Then it sounds like you're asking specific terms in the Euler-Lagrange equation to each correspond to specific functional derivatives of a functional - as though you were really positing two separate equations rather than one. $\endgroup$– Michael EngelhardtCommented Nov 25, 2019 at 3:25
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1$\begingroup$ @liding you seem to have changed the question so the manipulation I suggested doesn't immediately work $\endgroup$– AlexArvanitakisCommented Nov 25, 2019 at 14:06
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1$\begingroup$ @AlexArvanitakis Well, if you admit sufficiently fancy functions in your potential ... $V(u)=u^2 /2 - u\ln (1+e^u ) - Li_2 (-e^u ) $ ... $\endgroup$– Michael EngelhardtCommented Nov 25, 2019 at 14:53
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