We study nonlinear elliptic partial differential equations like \[ \left \{ \begin {array}{ll} - \Delta u = g (x, u) & \text {in } \Omega \text {,}\\ u = 0 & \text {on } \partial \Omega \end {array} \right . \] via variational and topological methods. Solutions of the above equation are critical points of the functional \(\Phi : H_0^1 (\Omega ) \rightarrow \mathbb {R}\) defined by \[ \Phi (u) = \frac {1}{2} \int _{\Omega } | \nabla u |^2\, \mathrm {d} x - \int _{\Omega } G (x, u)\, \mathrm {d}x \text {,} \] where \(G (x, t) = \int _0^t g (x, \cdot )\), \(H_0^1 (\Omega )\) is one of the Sobolev spaces. Using functional analytic and topological methods, critical point theory provides estimate of the number of critical points for a given functional.
Therefore, to work in this field, in addition to Real Analysis, one needs to know some Functional Analysis, Partial Differential Equations, and Topology. Differential Geometry is closely related to the field. In what follows, I mention some elementary books (thin, and available from the links given) on these subjects.
C. Clason, Introduction to functional analysis, Compact Textbooks in Mathematics, Birkhäuser/Springer, Cham, [2020] ©2020.
L. Godinho, J. Natário, An introduction to Riemannian geometry, Universitext, Springer, Cham, 2014. With applications to mechanics and relativity.
A. Valli, A compact course on linear PDEs, Unitext, vol. 126, Springer, Cham, [2020] ©2020. La Matematica per il 3+2.
S. Kesavan, Nonlinear functional analysis: a first course, Texts and Readings in Mathematics, vol. 28, Springer, Singapore; Hindustan Book Agency, New Delhi, 2nd ed., [2022] ©2022.
S. H. Weintraub, Fundamentals of algebraic topology, Graduate Texts in Mathematics, vol. 270, Springer, New York, 2014.