Morse Index of Solutions of Nonlinear Elliptic Equations

The book is a thorough, concise and elegant treatment of semilinear elliptic equations of the form \( \Delta u= f(x, u)\) with Dirichlet or mixed boundary conditions with an emphasis on the variational characterization of the eigenvalues that are needed for the Morse index of solutions. The content is appealing to graduate students or beginning researchers because of the inclusion of fundamental topics from Sobolev spaces, weak and strong maximum principles, and the classical eigenvalue theory for boundary value problems involving elliptic operators. Several important theorems that are spread in research articles or other books are put together in a single volume in the context of the elliptic theory. The methodology involves applications of Morse index (computation or estimation of it) based on kinds of solutions that are investigated.  Functions defined on cylindrically symmetric domains are considered. An emphasis is on the variational characterization of eigenvalues, which in fact is crucial to the study of the Morse index. Also included are linear and semilinear elliptic systems with Dirichlet conditions describing the related maximum principles and spectral theory. Nonetheless, the book does not have exercises and it also lacks examples to illustrate the theories. This may have happened because the reader is expected to have some mathematical maturity coming from the intrinsic interests in the subject. 

 

An entire chapter is devoted to the classical Morse theory for functions on finite-dimensional manifolds as well as its extension to the infinite dimensional case. Critical groups, Morse lemma, and Morse inequalities are elegantly presented. Morse index of mountain pass critical points is also estimated. Bifurcation from radial solutions of Lane-Emden Dirichlet problem in an annulus is included.

 

Connections between the Morse index of solutions of semilinear elliptic equations and their symmetry properties are given. Under some convexity conditions on the nonlinearity, the solutions which have Morse index not exceeding the dimension of the space are shown to have an axially symmetric property. Although these results exist in the literature, a unified framework simplifying proofs and adding recent developments for nonlinear mixed boundary conditions is presented in this book.   As some applications, the famous symmetry theorem of Gidas, Ni and Nirenberg is given by the method of moving planes, and counterexamples to the radial symmetry are given when the ball is replaced by an annulus or the nonlinearity does not satisfy the hypotheses of the theorem.

 

Symmetry results for solutions of elliptic cooperative systems on balls or annuli are obtained by using Morse index bounds.  More simplified versions of proofs than those found in the literature are given. The book concludes with some nonexistence results for scalar case in unbounded domains. 

 

In summary, the book is a great resource for both a graduate student and beginning researcher for an excursion into Morse index in finite and infinite dimensions and its applications to semilinear elliptic equations and systems. 

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Dhruba R. Adhikari is Associate Professor of Mathematics at Kennesaw State University.  His web page is http://facultyweb.kennesaw.edu/dadhikar/