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This textbook presents the essential parts of the modern theory of nonlinear partial differential equations, including the calculus of variations.

After a short review of results in real and functional analysis, the author introduces the main mathematical techniques for solving both semilinear and quasilinear elliptic PDEs, and the associated boundary value problems. Key topics include infinite dimensional fixed point methods, the Galerkin method, the maximum principle, elliptic regularity, and the calculus of variations.

Aimed at graduate students and researchers, this textbook contains numerous examples and exercises and provides several comments and suggestions for further study.

Nonlinear Elliptic Partial Differential Equations: An Introduction

1 A brief review of real and functional analysis.- 2 Fixed point theorems and applications.- 3 Superposition operators.- 4 The Galerkin method.- 5 The maximum principle, elliptic regularity, and applications.- 6 Calculus of variations and quasilinear problems.- 7 Calculus of variations and critical points.- 8 Monotone operators and variational inequalities.- References.- Index.