# Proving that a functional J satisfies the Palais Smale condition

I am reading Le Dret's book on Nonlinear Elliptic Partial Differential Equations.

On chapter 7 (page 209) I am trying to prove that the functional $$J(u) = \frac{1}{2}\int \lVert \nabla u\rVert^2 - \int G(u)$$, that arise from the study of the boundary-value problem $$-\Delta u = G'(u)\doteq g(u)$$ in $$H_0^1(\Omega)$$, satisfies the Palais Smale condition if $$g$$ has some growth property:

My issue is the following: Here the author says to conclude that $$u_n$$ is bounded just like on a previous proposition

If you go to the previous proposition this is what he is referring to:

My problem is: On the proposition 7.3 we have the equality $$DJ(u_n)u_n = (p+1)J(u_n) - \frac{p-1}{2}\int \lVert \nabla u_n\rVert^2$$ and then you may use the norm inequality for $$DJ(u_n)$$. But in the lemma 7.5, the one I'm trying to prove, we only have that $$DJ(u_n)u_n\leq C m(\Omega)+\theta J(u_n) +(1-\frac{\theta}{2})\int \lVert \nabla u_n\rVert^2$$. I am not sure how to conclude that $$u_n$$ is bounded from this.

Could someone help me in this passage?

Since $$\theta>2$$, one gets from the upper bound of $$DJ(u_n)u_n$$ $$(\frac\theta2-1) \|\nabla u\|_{L^2}^2 \le Cm(\Omega) + \theta J(u_n) - DJ(u_n)u_n.$$ Using the estimate as in Prop 7.3 $$(\frac\theta2-1) \|\nabla u\|_{L^2}^2 \le Cm(\Omega) + \theta J(u_n) +c\|DJ(u_n)\|_{H^{-1}}^2 +\frac12(\frac\theta2-1) \|\nabla u\|_{L^2}^2$$ with $$c = \frac12 (\frac\theta2-1)^{-1}$$.
• Thanks for the tip, I think I get most of it now. My only problem is: why does he references prop 7. 3? Didn't you use only something along the lines of $\lvert ab\rvert\leq \frac{1}{2k} a^2 + \frac{k}{2} b^2$ ? How is this any similar as 7.3? He is referring to 7.3 only for the norm of a linear functional inequality ? it seems odd to me Oct 22 '20 at 14:14
• Just realised that my question was really dumb, I didn't realize that the $\int \lVert \nabla u_n\rVert^2$ had a negative sign, thank you though Oct 22 '20 at 18:15
The solution is actually very simple:/, I did not realize that the term $$\int \lVert \nabla u_n\rVert^2$$ had a negative sign on the inequality that I have on lemma 7.5.