Consider the following quasilinear elliptic equation
$$\nabla\_x (A(x,u(x),\nabla\_x u(x))) + f(u,x) = 0 $$
on a bounded domain $\Omega$, augmented with homogeneous Dirichlet boundary data: $$u|\_{\partial \Omega} = 0.$$


* Where can I find references on this kinds of problems (in particular about a variational approach to solve them)?
* What's the correct formulation of a "uniform ellipticity"-type condition that gives existence and uniqueness for this problem?



Quasilinear elliptic problem: Ellipticity-type conditions
Here is a basic result. For a detailed proof, see Theorem 5.4 in [2].



> 
> **Theorem.** If $\Omega\subset\mathbb{R}^n$ is a bounded Lipschitz domain and $$
>  F=F(x,u,\xi):\Omega\times\mathbb{R}\times\mathbb{R}^n\to\mathbb{R} $$ 
>  satisfies
> 
> 
> * $F$ and $\nabla\_\xi F$ are continuous,
> * $\xi\mapsto F(x,u,\xi)$ is convex,
> * $F(x,u,\xi)\geq c|\xi|^p + a(x)$, $1<p<\infty$, $a\in L^1(\Omega)$,
> 
> 
> then for any $w\in W^{1,p}(\Omega)$ the functional $$ I(u)=\int\_\Omega
>  F(x,u,\nabla u)\,dx $$ attains minimum in the class $$
>  W^{1,p}\_w(\Omega)=\{ u\in W^{1,p}(\Omega):\, u-w\in
>  W^{1,p}\_0(\Omega)\}. $$
> 
> 
> 


Under some additional assumptions about $F$ this minimizer solves the elliptic equation
$$
(\*)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ div A(x,u,\nabla u)=B(x,u,\nabla u),
\quad
u-w\in W^{1,p}\_0(\Omega), 
$$
where 
$$
A(x,u,\xi)=\nabla\_\xi F(x,u,\xi),
\quad
B=F'\_u(x,u,\xi).
$$
Thus one of the method of solving (\*) is trough the associated variational problem.


Note also that convexity of $\xi\mapsto F$ implies that
$$
(\*\*)\ \ \ \ \ \ \ \ \ \ \langle A(x,u,\xi)-A(x,u,\eta),\xi-\eta\rangle\geq 0
$$
which is known as the monotonicity condition so another method of solving equation (\*) satisfying (\*\*) is through the Minty-Browder theorem. The monotonicity condition is a sort of ellipticity.


For an elementary, but detailed presentation of basic variational and non-variational techniques I highly recommend:


**[1] L. C. Evans,** *Partial differential equations.* Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010


Anther elementary presentation is given in:


**[2] P. Hajlasz,**
[*Non-linear elliptic partial differential equations.*](http://www.pitt.edu/~hajlasz/Notatki/Corlona%20lectures.pdf). Unpublished lecture notes.


Other books that develop material in depth are:


**[3] B. Dacorogna,** *Direct methods in the calculus of variations.* Second edition. Applied Mathematical Sciences, 78. Springer, New York, 2008.


**[4] M. Giaquinta,** *Introduction to regularity theory for nonlinear elliptic systems.* Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993. 
(Great book aut a lot of typos.)


**[5] M. Giaquinta, L. Martinazzi,** *An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs*. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 2. Edizioni della Normale, Pisa, 2005.


**[6] E. Giusti,** *Direct Methods in the Calculus of Variations.*
 World Scientific Publishing Co., Inc., River Edge, NJ, 2003.


