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Singularly perturbed methods for nonlinear elliptic problems. (English) Zbl 1465.35003

Cambridge Studies in Advanced Mathematics 191. Cambridge: Cambridge University Press (ISBN 978-1-108-83683-8/hbk; 978-1-108-87263-8/ebook). ix, 252 p. (2021).
This book concerns variational methods for non-compact elliptic problems, i.e. for semilinear elliptic PDEs either on the whole space, but with subcritical growth, or on a bounded domain, but with critical growth and homogeneous Dirichlet boundary conditions.The authors explain the main ideas by investigating two typical examples, the nonlinear Schrödinger problem \[ -\Delta u+V(x)u=u^p \text{ in } \mathbb{R}^N \] and the Brezis-Nirenberg problem \[ -\Delta u-\lambda u=u^{2^*-1} \text{ in } \Omega, \; u=0 \text{ on } \partial \Omega. \] Chapter 1 is devoted to the questions how, in non-compact elliptic problems, the compactness of a minimizing sequence can be recovered (energy constraints, mountain pass lemma) and how the loss of compactness is related to concentration phenomena (appearance of bumbs, bubbles, peaks). The Chapters 2 and 3 concern the blow-up behavior of nontrivial solutions to the Brezis-Nirenberg problem for \(\lambda \downarrow 0\) and to the singularly perturbed problem \[ -\varepsilon^2 \Delta u+V(x)u=u^p \text{ in } \mathbb{R}^N \] for \(\varepsilon \downarrow 0\). Existence and local uniqueness of concentrating solutions (with separated peaks as well as with clustering peaks) are shown by means of Pohozaev identities and the Lyapunov-Schmidt procedure. In Chapters 4 and 5 the authors construct infinitely many solutions to the nonlinear Schrödinger problem and the Brezis-Nirenberg problem.
All chapters include subchapters “Further results and comments”, where other related problems are briefly discussed and where corresponding references are given. Finally, in an appendix some background is presented (basic estimates, Pohozaev identities, Sobolev spaces, fundamental estimates for elliptic PDEs, Kelvin transformation, Green’s function estimates).
To summarize: This book presents in a very nice and self-contained manner the main methods to find (or to construct) solutions, which exhibit a concentration property, to non-compact elliptic problems. One question about terminology remains: Why these methods are called “singularly perturbed”?

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35B25 Singular perturbations in context of PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J61 Semilinear elliptic equations
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