Dtyjlui

The following is given:
Let $Omegasubset mathbb{R}^n$ be a bounded, connected open set with Lipschitz boundary. Let $fin C^0(overline{Omega}times mathbb{R}times mathbb{R}^n)$, $f=f(x,u,xi)$, satisfy

H1) $ximapsto f(x,u,zeta)$ is convex for every $(x,u)in overline{Omega}times mathbb{R}$

H2)there exist $p>qgeq1$ and $alpha_1>0, alpha_2, alpha_3in mathbb{R}$ such that $$f(x,u,xi)geqalpha_1|xi|^p+alpha_2|u|^q+alpha_3 forall(x,u,xi)inoverline{Omega}times mathbb{R}timesmathbb{R}^n$$
Set $alpha in mathbb{R}$ and $X={uin W^{1,p}(Omega)| frac{1}{|Omega|}int_{Omega}u dx=alpha}$

I intent to show the existence of a minimizer for
$$(P) inf{I(u)=int_{Omega}f(x,u,nabla u)dx|uin X}=m$$
by using minimizing sequences. I could already proof the lower semicontinuity of I(u). For the minimizing sequences I got the following:

Let $overline{u}equiv alpha$ on $Omega$. since $Omega$ is bounded I got $I({overline{u}})<infty$, so I can assume the existence of a minimizing sequence $(u_n)_{nin N}$. I first want to proof $$||u_n||_{W^{1,p}}<c, c constant$$ so i can conclude the existence of a convrgent subsequence.
Here I run into my first problem:

Since I got a minimizing sequence I got for $n$ large enought
$$m+1geq I(u_n)underset{H2}{geq} alpha_1 ||nabla u||^p_{L^p}+alpha_2||u||^q_{L^q}+alpha_3$$
Using the Hoelder inequility I get $||u||^q_{L^q}leq (const.)||u||^q_{L^p}$, but since $alpha_2$ can be either positiv or negativ I cant actually use that. I got the same problem when I tried to use Sobolev embedding theorem and I can't use Poincare inequility since I don't necessarily have $u_nin W^{1,p}_0$.

My second problem is showing that the limit of said subsequence is actually in X.

Disclaimer: As pointed out in the comment section, I am giving an answer for $alpha_2 geq 0$. While writing the answer, I also noted that my solution requires $Omega$ to be connected.

Step 1: $I(cdot)$ is bounded below

For any $u in X$, we have
$$
I(u)geq alpha_1||nabla u||_{L^p}^p+alpha_2||u||_{L^q}^q+alpha_3geq alpha_3 >- infty
$$

You already pointed out why $X$ is non-empty.

Step 2: Dealing with the minimizing sequence

So our functional is bounded below. Hence $m=inf_{u in X} I(u)$ exists. There exists a sequence $ { u_n }_{n in mathbb{N}} subset X$ such that $I(u_n)to m$.

We can assume that the sequnce statsifies $M geq I(u_n)$ for some $M>0$. Then, we can estimate
$$
M geq I(u_n)geqalpha_1||nabla u_n||_{L^p}^p+alpha_2||u_n||_{L^q}^q+alpha_3geq alpha_1||nabla u_n||+alpha_3
$$
This is possible, since $alpha_2||u_n||_{L^q}^qgeq 0$ holds now.

Therefore, since $alpha_1 >0$:
$$
||nabla u_n||_{L^p}^p leq frac{M-alpha_3}{alpha_1}
$$
This gives us an upper bound on $||nabla u_n||_{L^p}$.
Since all of our functions have a fixed mean value (as a property of $X$), you can now use a Poincare inequality involving mean values:
$$
||u_n||_{L^p}=||u_n-alpha +alpha||_{L^p}leq ||u_n-alpha||_{L^p}+||alpha||_{L^p}leq
C_1||nabla u_n||_{L^p}+C_2
$$
A reference for this can be found, for example, in Evans PDE book. While writing the answer, I noted that we also need $Omega$ to be connected.

Combining this with the bound on the gradient, you get that
$$
||u_n||_{W^{1,p}}leq C
$$

Now we have a bounded sequnce in $W^{1,p}$, a reflexive Banach space. Now we can find a weakly convergent subsequence $u_k rightharpoonup w$, where we have $w in W^{1,p}$. It remains to show that $w in X$. But this is due to Mazurs theorem. The set $X$ is convex and closed (why?) with respect to $||cdot ||_{W^{1,p}}$ and hence weakly closed.

Step 3: Passing to the limit

You have already noted that this "uniform" convexity in $zeta$ and boundedness below implies weak lower semicontinuity. This implies:
$$
I(w)leq liminf_{k to infty}I(u_k)=m=inf_{u in X} I(u)
$$

This leads to the conclusion:
$$
I(w)=min_{u in X} I(u)
$$

A few notes:

1) You did not need the Sobolev embedding at any point.

2) I left in a few gaps, since it would be way too much to write. A lot of theorems/steps use for example the properties of $Omega$

3) The poincare inequality you were looking for was
$$
||u_n-frac{1}{|Omega|}int_{Omega}u_n(x)dx||_{L^p(Omega)}leq C||nabla u_n||_{L^p(Omega)}
$$

However, this needs the region $Omega$ to be connected. I did forget about this while thinking about your post.

I hope this answer is still helpful and gives you an idea on how to deal with those problems.

4) I did not need the term $alpha_2||u||_{L^q}^q$ at all. In fact, as $||u||_{L^q} leq C_3 ||u||_{L^p}leq C_4(||nabla u_n||_{L^p}+1)$ makes the term look like it does not contain any valueable information. Now you might see what I tried: If $alpha_2 < 0$, you could try to justify something like:
$$
alpha_2||u||_{L^q} geq alpha_2 C_4(||nabla u_n||_{L^p}+1)
$$
After some calculations, it will boil down to the sign of
$$
alpha_1+C_5 alpha_2
$$
This is basically impossible to predict, as $C_5=C_5(Omega,p,n,q)geq 0$.
Other techniques I know of were not applicable.

As a last and additional note for the $alpha_2 < 0$ condition: You would need that for every sequence such that $alpha_1 ||nabla u_n||_{L^p}^p+alpha_2||u_n||_{L^q}^q to - infty$ the functional stays bounded. Also, it all needs to be a sequence $X$, which means it has a fixed mean value.