# Would you help me to understand which result from Sobolev Spaces or PDE are being used here?

I've never studied PDE and I only studied a little about Sobolev Spaces by myself. I would like to understand why can I conclude that $$u_{m,n}$$ is a weak solution to the Dirichlet problem below. Is being used any theorem? Or is being used only a definition of a weak solution? or is it only a matter of developing calculations related to the previous inequality in 2.10? I also did not understand why there is an infimum before a maximum in (2.10), would be about any definition, or is being used any theorem?

P.S: I'm studying for my final project in my course however is very hard to understand many steps in the paper. Despite of this, I'm really enjoying everything I'm learning. Where (2.6), (2.7) and (2.8) are And, the functional $$J_{m,n}$$ being mentioned is • Try to calculate directional derivative first. Think of $\nabla$ as an ordinary linear operator.
– daw
Feb 10, 2022 at 12:31
• I wonder what is the derivative of $(v_+)^q$. Feb 10, 2022 at 16:11
• @ArcticChar nothing else than $q(v^+)^{q-1}$ provided $q\ge1$ (assuming $v^+=\max(v,0)$).
– daw
Feb 10, 2022 at 16:13
• Um.... I guess so. To the OP, you've got your $L_n'(v)$ wrong. Look like you just add $\varphi$ to the product and did not differentiate. Feb 10, 2022 at 16:17
• Much better than the first trial. The second term is missing a $\phi$
– daw
Feb 11, 2022 at 7:01

You are using two theorems. The first theorem is the Mountain Pass Theorem , which tells you that if the functional $$J_{m,n}$$ satisfies the hypotheses (2.6) and (2.7), then it admits a critical point, that is , a point $$u_{m,n}$$ such that the derivative of the functional $$J_{m,n}$$ computed at $$u_{m,n}$$ is zero. Then you are using another theorem that says that critical points of your functional are weak solutions of a partial differential equations, called the Euler Lagrange equations of your functional. You can find this derivation in the book by Evans “Partial differential equations”.