I would really appreciate to hear your insights or comments about the following problem: Consider the following non linear pde: let $\Omega$ be the unit square with vertices at (0,0),(1,0),(0,1) and (1,1) Consider $\triangle u - u^3 = 0$ and $ u = f(x,y)$ on $\partial \Omega$ where $f= \frac{1}{\epsilon}$ when $|x-\frac{1}{2}| \leq \frac{\epsilon}{2}$ and zero otherwise... what could be said about the energy as $\epsilon$ approaches zero?

If in particular there is a link with these kind of pdes please let me know...thanks



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