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Weak convergence methods for semilinear elliptic equations. (English) Zbl 1059.35038

Singapore: World Scientific (ISBN 981-02-4076-7/hbk). xii, 234 p. (1999).
Publisher’s description: This book deals with nonlinear boundary value problems for semilinear elliptic equations on unbounded domains with nonlinearities involving the subcritical Sobolev exponent. The variational problems investigated in the book originate in many branches of applied science. A typical example is the nonlinear Schrödinger equation which appears in mathematical modeling phenomena arising in nonlinear optics and plasma physics. Solutions to these problems are found as critical points of variational functionals. The main difficulty in examining the compactness of Palais-Smale sequences arises from the fact that the Sobolev compact embedding theorems are no longer true on unbounded domains. In this book we develop the concentration-compactness principle at infinity, which is used to obtain the relative compactness of minimizing sequences. This tool, combined with some basic methods from the Lusternik-Schnirelman theory of critical points, is to investigate the existence of positive, symmetric and nodal solutions. The book also emphasizes the effect of the graph topology of coefficients on the existence of multiple solutions.

MSC:

35J60 Nonlinear elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J10 Schrödinger operator, Schrödinger equation
35J20 Variational methods for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
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