Consider the following elliptic PDE: $$ \Delta u=f(u), $$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have a rough introduction? In particular, I am interested in applying variation methods (Calculus of variations) to investigate the existence problem for this PDE.
2026-03-28 05:45:51.1774676751
Elliptic partial differential equations
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I'd start with Evans, "Partial Differential Equations." A good reference for general second-order elliptic equations is Gilbarg and Trudinger, "Elliptic Partial Differential Equations of Second Order."