How to solve Elliptic PDE

I have been researching places like Wikipedia and have read research papers, and still do not know how to solve second order Elliptic PDEs. I know how to solve first order elliptic PDEs, and by my understanding, elliptic PDEs are of the form $$Au_{xx} + 2Bu_{xy}+ Cu_{yy} + Du_x+ Eu_y+ Fu + G = 0$$ How would I go around solving a second order elliptic PDE, or are there any resources I should look at to learn about this topic?

• Important aspects of an elliptic PDE include the boundedness of the region, the smoothness of the boundary, and the associated boundary conditions. For key, fundamental theory you can't do too much better than Kellogg's "Foundations of Potential Theory," followed perhaps by Hackbusch's "Elliptic Differential Equations," but naturally that's a matter of taste. Peter Knabner has a nice book on both elliptic and parabolic systems. Jul 3, 2021 at 23:29
• This question is probably too broad for a site like this, since there is no short answer. A good place to start would be any basic PDE textbook (see math.stackexchange.com/questions/2827/…). Jul 4, 2021 at 6:08
• By the way, what are “first order elliptic PDEs”, if I may ask? Jul 4, 2021 at 6:10

The first thing to note is that the form you wrote down isn't necessarily elliptical. It's only defined as elliptical if

$$B^2 - A C < 0.$$

From there, one thing you can do is either take a Fourier transform or, in a similar vein, assume a solution of the form

$$u(x,y)=\exp(i k_x x + i k_y y),$$

where $$k_x, k_y$$ could be complex. The elliptic equation would then give you a relationship between $$k_x, k_y$$ that would have to be satisfied. By linearity, you can compose general solutions as linear combinations of solutions of this form. Also by linearity, if you want real solutions and you have real coefficients you can just take the real part of your solution.