##
**Singular elliptic problems: Bifurcation and asymptotic analysis.**
*(English)*
Zbl 1159.35030

Oxford Lecture Series in Mathematics and Its Applications 37. Oxford: Oxford University Press (ISBN 978-0-19-533472-2/hbk). xvi, 298 p. (2008).

The book is an introduction to qualitative analysis of nonlinear singular stationary problems and fills a gap in the literature concerning this topic. The theory is explained on basic model examples of elliptic problems with different types of singularities. The problems are studied on domains in the Euclidean space, even if a generalization to differentiable manifolds would be possible. Mainly classical solutions are considered.

In the first part of the book, some basic methods and results used in the next chapters are summarized (sub- and super-solution method, maximum and comparison principles, existence and uniqueness results for elliptic boundary value problems).

The second part is devoted to the study of singular solutions of logistic-type equations. First, blow-up boundary behavior of solutions of such problems on smooth domains is considered (Chapter 2) and then entire solutions on the whole space blowing-up at infinity are discussed (Chapter 3).

Hence, the singular character of the problems considered here is given by the prescribed behavior of solutions at the boundary, not by a singularity in nonlinear terms.

The third part of the book is concerned with qualitative analysis of elliptic problems involving mainly singular nonlinearities. Positive solutions of problems of the type \(-\Delta u=\Phi(x,u,\lambda)\) usually with Dirichlet boundary conditions are studied in smooth domains in \(\mathbb R^N\), where \(\Phi\) is allowed to be unbounded near the origin in the second variable, \(\lambda\) is a parameter. Several subclasses of problems of this type are considered in Chapter 4. For the equations \(-\Delta u=\lambda f(u)\), a description of existence/nonexistence and stability of solutions for parameters \(\lambda\) from different intervals is given and asymptotic properties for \(\lambda\) going to infinity are shown.

Continuous right hand sides as well as those having a singularity are discussed. The monotone case (Chapter 5) as well as non-monotone case (Chapter 6) are studied. Properties of singular elliptic problems in the presence of smooth nonlinearities with a super-linear growth at infinity are demonstrated on the model example \(-\Delta u=u^{-\alpha} + \lambda u^p\) (Chapter 7). Existence/nonexistence of (this time weak) solutions in the dependence on parameter and their regularity is studied, bifurcation and asymptotic behavior as \(p\to 1\) is described.

The stability of positive solutions of a singular problem of the type \(-\Delta u=f(x,u)\) is studied (Chapter 8). The influence of a convection term in singular elliptic problems is shown on the equation

\[ -\Delta u = p(d(x))g(u) + \lambda |\nabla u|^a + \mu f(x,u), \]

where the function \(g\) has a singularity at the origin, \(p\) is a possibly singular weight, \(d(x)=\text{dist}(x, \partial \Omega)\) (Chapter 9). The existence/nonexistence of positive solutions and multiplicity results depending on the parameters \(\lambda\), \(\mu\) are given. The last chapter is devoted to a class of elliptic systems with singular right hand sides which includes a stationary problem corresponding to the Gierer-Meinhardt system having applications in biology. Existence/nonexistence, unicity and properties of solutions are studied.

The book is completed by four appendices devoted to basic tools which are used - spectral theory for differential operators, the implicit function theorem, the Eckeland’s variational principle and the mountain pass theorem. Hence, the book is self contained, only a standard knowledge of the PDE’s is sufficient for the study. It can be recommended to advanced graduate students as well as to researchers in pure and applied mathematics and physics.

In the first part of the book, some basic methods and results used in the next chapters are summarized (sub- and super-solution method, maximum and comparison principles, existence and uniqueness results for elliptic boundary value problems).

The second part is devoted to the study of singular solutions of logistic-type equations. First, blow-up boundary behavior of solutions of such problems on smooth domains is considered (Chapter 2) and then entire solutions on the whole space blowing-up at infinity are discussed (Chapter 3).

Hence, the singular character of the problems considered here is given by the prescribed behavior of solutions at the boundary, not by a singularity in nonlinear terms.

The third part of the book is concerned with qualitative analysis of elliptic problems involving mainly singular nonlinearities. Positive solutions of problems of the type \(-\Delta u=\Phi(x,u,\lambda)\) usually with Dirichlet boundary conditions are studied in smooth domains in \(\mathbb R^N\), where \(\Phi\) is allowed to be unbounded near the origin in the second variable, \(\lambda\) is a parameter. Several subclasses of problems of this type are considered in Chapter 4. For the equations \(-\Delta u=\lambda f(u)\), a description of existence/nonexistence and stability of solutions for parameters \(\lambda\) from different intervals is given and asymptotic properties for \(\lambda\) going to infinity are shown.

Continuous right hand sides as well as those having a singularity are discussed. The monotone case (Chapter 5) as well as non-monotone case (Chapter 6) are studied. Properties of singular elliptic problems in the presence of smooth nonlinearities with a super-linear growth at infinity are demonstrated on the model example \(-\Delta u=u^{-\alpha} + \lambda u^p\) (Chapter 7). Existence/nonexistence of (this time weak) solutions in the dependence on parameter and their regularity is studied, bifurcation and asymptotic behavior as \(p\to 1\) is described.

The stability of positive solutions of a singular problem of the type \(-\Delta u=f(x,u)\) is studied (Chapter 8). The influence of a convection term in singular elliptic problems is shown on the equation

\[ -\Delta u = p(d(x))g(u) + \lambda |\nabla u|^a + \mu f(x,u), \]

where the function \(g\) has a singularity at the origin, \(p\) is a possibly singular weight, \(d(x)=\text{dist}(x, \partial \Omega)\) (Chapter 9). The existence/nonexistence of positive solutions and multiplicity results depending on the parameters \(\lambda\), \(\mu\) are given. The last chapter is devoted to a class of elliptic systems with singular right hand sides which includes a stationary problem corresponding to the Gierer-Meinhardt system having applications in biology. Existence/nonexistence, unicity and properties of solutions are studied.

The book is completed by four appendices devoted to basic tools which are used - spectral theory for differential operators, the implicit function theorem, the Eckeland’s variational principle and the mountain pass theorem. Hence, the book is self contained, only a standard knowledge of the PDE’s is sufficient for the study. It can be recommended to advanced graduate students as well as to researchers in pure and applied mathematics and physics.

Reviewer: Milan Kučera (Praha)

### MSC:

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35J70 | Degenerate elliptic equations |

35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |

35B32 | Bifurcations in context of PDEs |

35B35 | Stability in context of PDEs |

35B65 | Smoothness and regularity of solutions to PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35J20 | Variational methods for second-order elliptic equations |