The use of variational principles has a long and fruitful history in mathematics and physics, both in solving problems and shaping theories, and it has been introduced recently in economics. The corresponding literature is enormous and several monographs are already classical. The present book Variational Principles in Mathematical Physics, Geometry, and Economics, by Kristály, Rӑdulescu and Varga, is original in several ways.

In Part I, devoted to variational principles in mathematical physics, unavoidable classical topics such as the Ekeland variational principle, the mountain pass lemma, and the Ljusternik–Schnirelmann category, are supplemented with more recent methods and results of Ricceri, Brezis–Nirenberg, Szulkin, and Pohozaev. The chosen applications cover variational inequalities on unbounded strips and for area-type functionals, nonlinear eigenvalue problems for quasilinear elliptic equations, and a substantial study of systems of elliptic partial differential equations. These are challenging topics of growing importance, with many applications in natural and human sciences, such as demography.

Part II demonstrates the importance of variational problems in geometry. Classical questions concerning geodesics or minimal surfaces are not considered, but instead the authors concentrate on a less standard problem, namely the transformation of classical questions related to the Emden–Fowler equation into problems defined on some four-dimensional sphere. The combination of the calculus of variations with group theory provides interesting results. The case of equations with critical exponents, which is of special importance in geometrical problems since Yamabe's work, is also treated.