A variational problem - some guidance 
This is a problem I'm thinking about, to learn some more advanced calculus of variations on my own. I would appreciate some help, or a solution, just to have a sample to compare in the future.

Let


*

*$\Omega$ be a bounded open subset of $\mathbf{R}^N$, $N\geq3$;

*$a(x)$ an almost everywhere bounded measurable function on $\Omega$
($0<\alpha\leq a(x)\leq\beta$, with $\alpha,\beta\in\mathbf{R}_{>0}$, for almost all $x\in\Omega$);

*$1<p<N$;

*$F(x)\in(L^m(\Omega))^N$.


I'm interested in the existence of weak solutions $u\in W_0^{1,p}(\Omega)$ of the Dirichlet problem:
$$
(P)\ \ \ \forall v\in W_0^{1,p}(\Omega)\ \ \int_{\Omega} a(x)|\nabla u|^{p-2}\cdot\nabla u\cdot\nabla v=\int_{\Omega}F\cdot\nabla v,$$
their boundedness and uniqueness. 
 A: This seems a basic exercise in calculus of variations, so the standard answer is that you should think about it yourself a bit harder, and perhaps ask some questions to somebody at your department.
It's likely you won't get an answer from experts, then.
Since I'm no expert and it's been quite some time since the last time I thought about this kind of questions, I'm happy to give it a try, and your exercise, this once, can become to carefully proof-check what's here below.
I'll use your notation, and the rest of the notation below should be standard.
The main idea is to find a solution as a critical point of some appropriate functional on $W_0^{1,p}(\Omega)$, and the problem seems rigged on purpose for you to define 
$$J(u):= \frac{1}{p}\int_{\Omega} a(x)|\nabla u|^{p}-\int_{\Omega}b(x,\nabla u),$$
where $b(x,\nabla u)=F(x)\cdot\nabla u$, and, morally, find those points $u\in W_0^{1,p}(\Omega)$ at which the "prime derivative" of $J$ is zero.
One has to only be careful about making precise sense of this. The remaining questions all turn into questions about $J$, and you have several tools at your disposal to answer them.
Below, I won't be too pedantic (but you should make sure you understand where identities and inequalities are meant "almost everywhere" and where they aren't).
1. The problem is well-posed
Indeed, let's compute the derivative of $J$ in the sense of Gateaux, with respect to $u\in W_0^{1,p}(\Omega)$: pick an arbitrary direction $0\neq v\in W_0^{1,p}(\Omega)$, and compute:
$$
\lim_{t\to 0}\frac{J(u+tv)-J(v)}{tv}= \frac{\text{d}}{\text{d} t}J(v+tu)\vert_{t=0}\\
      = \frac{\text{d}}{\text{d} t}\left\{\frac{1}{p}\int_{\Omega}a(x)|\nabla (u+tv)|^p-\int_{\Omega}b(x,\nabla(u+tv))\right\}_{t=0}\\
  = \int_{\Omega}a(x)|\nabla u|^{p-2}\nabla u\cdot\nabla v-\frac{\text{d}}{\text{d} t}\int_{\Omega}F(x)\cdot(\nabla u+t\nabla v)\\
  = \int_{\Omega}a(x)|\nabla u|^{p-2}\nabla u\cdot\nabla v-\int_{\Omega}F(x)\nabla v
$$
Likewise as for the functional $J(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^p$, the derivative in the sense of Gateaux of $J$ is defined along all directions $0\neq v\in W_0^{1,p}(\Omega)$, compatibly, whence $J$ is differentiable in the sense of Fréchet, and its derivative satisfies $J'(u)\in W^{-1,p'}(\Omega)$.
Being $u,v\in W_0^{1,p}(\Omega)$, we have $|\nabla u|^{p-2}\nabla u\cdot\nabla v\in L^1(\Omega)$, and since $a$ is almost everywhere bounded, ill-posedness issues may only arise from the source term.
For this not to be the case, it is enough to have $F(x)\cdot\nabla v\in L^1(\Omega)$ for all $v\in W_0^{1,p}(\Omega)$.
We use
$$-\text{div}(F)v=-\text{div}(F\cdot v)+F\cdot\nabla v$$
and have that $F\cdot\nabla v\in L^1(\Omega)$ for all $v\in W_0^{1,2}(\Omega)$ if and only if $\text{div}(F)v\in L^1(\Omega)$ for all $v\in W_0^{1,p}(\Omega)$, if $\text{div}(F)\in L^2(\Omega)$, and we are through.
For the sake of clarity, let us make the Dirichlet problem for which $(P)$ is the weak formulation, explicit. 
$(P)$ is the weak formulation of:
$$(D)\ \ \ \left\{\begin{array}{cc}
-\text{div}(a(x)|\nabla u|^{p-2}\nabla u)=-\text{div}(F) & \text{in}\ \Omega\\
u=0 & \text{over}\ \partial\Omega
\end{array}\right.
$$
By Rellich-Kondrachev, $$W_0^{1,p}(\Omega)\subset L^q(\Omega)$$ is a compact immersion for all $1\leq q<p^*$, whence, since $\Omega$ is bounded, if $\text{div}(F)\in L^s(\Omega)$, then $\text{div}(F)v\in L^1(\Omega)$ if (and only if) $v\in L^{s'}(\Omega)$ for all $v\in W_0^{1,p}(\Omega)$. We know this is the case for:
$$1\leq s'<p^*$$
Explicitly:
$$1\leq\frac{s}{s-1}< p^*$$
whence $s> \frac{p^*}{p^*-1}=p_*$. Finally, if $\text{div}(F)\in L^s(\Omega)$, then $F\cdot\nabla v\in L^1(\Omega)$, whence, since $\nabla v\in (L^{s'}(\Omega))^N$,
we have $F\in (L^s(\Omega))^N$, that is, $s=m$. It follows the question is indeed well-posed for $m> p_*$.
2. Existence by minimization
In order to determine existence of solutions to the Euler equation:
$$\forall v\in W_0^{1,p}(\Omega)\ \ \ J'(u)(v)=0$$
by minimizing the functional $J:W_0^{1,p}(\Omega)\to\mathbf{R}$, we'll use a well known minimization Theorem of Weierstrass asserting existence of a minimum for coercive weakly lower semi-continuous operators on reflexive Banach spaces.
Since $W_0^{1,p}(\Omega)$ is Banach and reflexive, we need only prove:
(a) $J$ is weakly lower semi-continuous;
(b) $J$ is coercive.
Proof of (a). We call $g(x,s,\xi):=\frac{1}{p}a(x)|\xi|^p$, a measurable function of $x\in\Omega$, costante (hence continuous) in $s$, continuous, and in fact differentiable, in $\xi$, for almost all $x\in\Omega$. A Theorem of De Giorgi would already yield the contention, but we give a direct argument below anyway.
Let $v_n\to v$ weakly in $W_0^{1,p}(\Omega)$. We now use the assumptions on $a$, hence on $g$. Since $g$ is differentiable as a function of $\xi$, we have:
$$g(x,s,\xi)=g(x,s,\eta)+\partial_{\xi}g(x,s,\eta)\cdot(\xi-\eta)+O(|\xi-\eta|^2)\geq g(x,s,\eta)+\partial_{\xi}g(x,s,\eta)\cdot(\xi-\eta)$$
Upon integrating both sides, we have:
$$\frac{1}{p}\int_{\Omega}a(x)|\nabla v_n|^p\geq\frac{1}{p}\int_{\Omega}a(x)|\nabla v|^p+\int_{\Omega}\partial_{\xi}g(x,s,\nabla v)\cdot(\nabla v_n-\nabla v)$$
$\nabla v_n\to\nabla v$ weakly in $(L^p(\Omega))^N$ by assumption, and:
$$\partial_{\xi}a(x)|\nabla v|^p\leq p\beta|\nabla v|^{p-1}\in L^{p'}(\Omega),$$
whence, by the Bounded Convergence Theorem:
$$\int_{\Omega}\partial_{\xi}\{a(x)|\nabla v|^p\}\cdot(\nabla v_n-\nabla v)\to 0.$$
We deduce:
$$\liminf_n\int_{\Omega}a(x)|\nabla v_n|^p\geq\int_{\Omega}a(x)|\nabla v|^p.$$
Since the term $-\int_{\Omega}F(x)\cdot\nabla u$ is continuous in $u$, the contention follows. QED
Proof of (b). This is immediate by the assumptions on $a$:
$$a(x,s)\geq\alpha>0,$$
which yields:
$$g(x,s,\xi)\geq \alpha|\xi|^p.$$
By the Hölder inequality, we have:
$$\int_{\Omega}|F(x)\cdot\nabla u|\leq\left(\int_{\Omega}|F(x)|^s\right)^{1/s}\cdot\left(\int_{\Omega}|\nabla u|^r\right)^{1/r},$$
and upon choosing $s=m$ and $r=m'$, we have:
$$\int_{\Omega}|F(x)\cdot\nabla u|\leq\||F|\|_m\cdot\left(\int_{\Omega}|\nabla u|^{m'}\right)^{1/m'}\leq\||F|\|_m\cdot \left(\int_{\Omega}|\nabla u|^{rt}\right)^{1/rt}\cdot|\Omega|^{\frac{t-1}{t}}$$
which is:
$$\int_{\Omega}|F(x)\cdot\nabla u|\leq\||F|\|_m\cdot\||\nabla|\|_p\cdot|\Omega|^{\frac{pm-p-m}{pm-p}}.$$
Note that the steps in the foregoing are indeed legitimate, as $m>p^*$. It follows:
$$|J(u)|\geq \frac{\alpha}{C_{p,\Omega}}\cdot \|u\|_{W_0^{1,p}(\Omega)}^p-c_p\||f|\|_m\cdot\|u\|_{W_0^{1,p}(\Omega)},$$
by the Poincaré inequality, whence $\alpha/C_{p,\Omega}$ and $c_p$ are constants, depending only on $p$ and $\Omega$. Coerciveness of $J$ follows. QED
A weak solution $u\in W_0^{1,p}(\Omega)$ to the problem $(P)$, therefore, exists, in the form of a critical point (a minimum, in fact) of the functional $J$.
3. Uniqueness
Assume $u_1$ and $u_2$ in $W_0^{1,p}(\Omega)$ are two solutions. We have:
$$J'(u_1)(v)=\int_{\Omega} a(x)|\nabla u_1|^{p-2}\cdot\nabla u_1\cdot\nabla v=\int_{\Omega}F\cdot\nabla v$$
$$J'(u_2)(v)=\int_{\Omega} a(x)|\nabla u_2|^{p-2}\cdot\nabla u_2\cdot\nabla v=\int_{\Omega}F\cdot\nabla v.$$
We choose $u_1-u_2$ as $v$. Upon subtracting on both sides, we get:
$$I_p:=\int_{\Omega}a(x)\left\{|\nabla u_1|^{p-2}\nabla u_1-|\nabla u_2|^{p-2}\nabla u_2\right\}\cdot\nabla(u_1-u_2)=0.$$
We treat the case $p\geq 2$, using the inequality:
$$(|t|^{p-2}t-|s|^{p-2}s)\geq c_p\cdot(t-s)^{p-1}$$
for a positive constant $c_p$ and all $s,t\in\mathbf{R}$, and using the assumption $a(x)\geq\alpha>0$ (for almost all $x\in\Omega$).
We get:
$$0=I_p\geq\alpha\cdot c_p\int_{\Omega}|\nabla u_1-\nabla u_2|^{p}\geq\alpha\cdot c_p\cdot C_{p,\Omega}\|u_1-u_2\|_{W_0^{1,p}(\Omega)}$$
from the Poincaré inequality again. It follows $u_1=u_2$ almost everywhere on $\Omega$.
The case $1<p<2$ is done with the uniform convexity inequality, and is left to you to think about.
4. Boundedness
The answer is: the solution is bounded for $m>\frac{Np}{Np-2N+p}.$
Upon defining $g(x,\xi)=a(x)|\xi|^{p-2}\xi$, we use $G_k(u)$, with $u$ a solution to $(P)$, as test function. We have, for $J'(u)(G_k(u))$,
$$\int_{\Omega_k}g(x,\nabla G_k(u))\cdot\nabla G_k(u)=\int_{\Omega}g(x,\nabla u)\cdot \nabla G_k(u)=\int_{\Omega}F\cdot\nabla G_k(u).$$
We have, on the left side, by ellipticity of $g$, a consequence of the assumptions on $a$, and on the right side, by the Hölder inequality with exponents $(r,s)$:
$$\alpha\int_{\Omega}|\nabla G_k(u)|^p\leq\left(\int_{\Omega}\{|F|\cdot \chi_{\Omega_k}\}^r\right)^{1/r}\cdot\left(\int_{\Omega}|\nabla G_k(u)|^s\right)^{1/s}$$
For the right side to make sense, we choose $r=m$, whence $s=m'$, upon observing that if we have $u\in W_0^{1,p}(\Omega)$, then also $G_k(u)\in W_0^{1,p}(\Omega)$ (check!).
We now estimate the last integral, again using the Hölder inequality with exponents $(h,k)$:
$$\left(\int_{\Omega_k}|\nabla G_k(u)|^{m'h}\right)^{1/m'h}\cdot|\Omega_k|^{\frac{1}{m'k}}.$$
For this to make sense, we choose $h=p/m'$. It follows $$m'k=\frac{m'p/m'}{h-1}=\frac{mp}{mp-p-m},$$
We obtain:
$$\alpha\||\nabla G_k(u)|\|_p\leq\||F|\|_m\cdot|\Omega_k|^{\frac{mp-p-m}{mp}}.$$
By Sobolev-Gagliardo-Niremberg, we get:
$$\alpha\cdot S_{p,\Omega}\left(\int_{\Omega}|G_k(u)|^{p^*}\right)^{1/p^*}\leq\||F|\|_{m}\cdot|\Omega_k|^{\frac{1}{m'}-\frac{1}{p}}.$$
We again estimate $\left(\int_{\Omega}|G_k(u)|^{p^*}\right)^{1/p^*}$ using the Holder ''backwards'':
$$\int_{\Omega}|G_k(u)|\leq\left(\int_{\Omega}|G_k(u)|^{p^*}\right)^{1/p^*}\cdot|\Omega_k|^{\frac{1}{p_*}}.$$
Tying everything together, we get:
$$\int_{\Omega}|G_k(u)|\leq C_{p,\Omega}(\alpha)\cdot\||F|\|_m\cdot|\Omega_k|^{\frac{m-1}{m}-\frac{1}{p}+\frac{Np-N+p}{Np}}$$
where $C_{p,\Omega}(\alpha):=(\alpha\cdot S_{p,\Omega})^{-1}$. $C_{p,\Omega}(\alpha)>0$, whence we are left to check:
$$\frac{m-1}{m}-\frac{1}{p}+\frac{Np-N+p}{Np}>1,$$
that is:
$$\frac{m-1}{m}>\frac{2N-p}{Np}.$$
We have:
$$m>\frac{Np}{Np-2N+p}.$$
We observe that for $p=2$ we have $m>N$, as expected. 
5. Summarizing
(P) is well posed for $m> p_*$, with unique solution $u\in W_0^{1,p}(\Omega)$, and with $u\in W_0^{1,p}(\Omega)\cap L^{\infty}(\Omega)$ for $$m>\frac{Np}{Np-2N+p}>p_*.$$
Remark
Since the operator:
$$A:W_0^{1,p}(\Omega)\longrightarrow W_0^{-1,p'}(\Omega)$$
assigned by $u\mapsto -\text{div}(a(x)|\nabla u|^{p-2}\nabla u)$ is well posed, and since for $m>p_*$, we have $-\text{div}(F)\in W_0^{-1,p'}(\Omega)$ (by Sobolev-Gagliardo-Niremberg).
existence of solutions also follows from the Leray-Lions Theorem.
I think another way to see existence is by the Ekeland principle, since we have $J\in C^1(W_0^{1,p}(\Omega),\mathbf{R})$, bounded below, and easily seen to satisfy the Palais-Smale condition.
Some references for the Thms used
I'll track them back when I have some time. Spare time is over now
