Variational methods for potential operator equations: with applications to nonlinear elliptic equations. (English) Zbl 1157.35338

de Gruyter Studies in Mathematics 24. Berlin: Walter de Gruyter (ISBN 3-11-015269-X/hbk). ix, 290 p. (1997).
This book is concerned with some aspects of variational calculus which have been used in the last 30 years in the theory of elliptic partial differential equations as well as Hamiltonian systems and nonlinear wave equations. Many books on the subject have appeared in the last few years, each one of them having its own point of view and its own emphasis [e.g. M. Struwe, “Variational Methods”, Springer Verlag, Berlin (1990; Zbl 0746.49010]; N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge Tracts in Math. and Math. Physics 107 (1993; Zbl 0790.58002); K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser, Boston (1993; Zbl 0779.58005)]. The main characteristic of the monograph of Chabrowski is that it deals with Euler-Lagrange equations of the form \(A(u)-\lambda B(u)=0\), where \(\lambda\) is a real number while \(A\), \(B\) are potential operators of degrees \(p\) and \(q\), respectively, with \(A\) being typically the homogeneous \(p\)-Laplacian while \(B\) is sometimes allowed to be non-homogeneous.
Chapter 1 deals with constrained minimization problems with a particular emphasis on the cases where they lead to global critical points as in the context when both \(A\) and \(B\) are both homogeneous and the case where the set of constraints is “artificial” á la Nehari. Chapter 2 deals with the standard Lyusternik-Shnirel’man theory associated to the Krasnosel’ski genus. Its applicability is illustrated by the variations that one obtains by considering the cases \(p<q\), \(p>q\) and \(p=q\). In Chapter 3, the homogeneity assumption on \(B\) is dropped in the constrained minimization problem and perturbation methods are developed to deal with the non-compact cases. Chapter 4 extends the results of Chapter 1 to the case where \(B\) is not homogeneous but satisfies a covariance condition with respect to a group of transformations. This allows the recovery of global critical points (i.e., no Lagrange multipliers). In Chapter 5, the non-homogeneous case in constrained minimization problems is again dealt with by studying the variations of the level sets that constitute the constraints. Schecter’s variations of the mountain pass theorem (MPT) are given. This continues in Chapter 6, while the “true” mountain pass theorem is proven in Chapter 7 in the case of non-differentiable but locally Lipschitz functionals. The Ghoussoub-Preiss MPT in the presence of duality is also included. Chapter 8 considers the subcritical case of P.-L. Lions’ concentration-compactness principle and two standard applications while Chapter 8 deals with the critical case.


35Jxx Elliptic equations and elliptic systems
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
47N20 Applications of operator theory to differential and integral equations
49J40 Variational inequalities