Singular solutions of semi-linear elliptic problems. (English) Zbl 1191.35131

Dávila, J.

Chipot, Michel (ed.), Handbook of differential equations: Stationary partial differential equations. Vol. VI. Amsterdam: Elsevier/North Holland (ISBN 978-0-444-53241-1/hbk). Handbook of Differential Equations, 83-176 (2008).

The present survey concerns singular solutions to semi-linear elliptic problems of the form
\[ \begin{cases} -\Delta u=\lambda g(u) & \text{in } \Omega\\ u=0 & \text{on } \partial\Omega, \end{cases} \]
where \(\Omega\) is a bounded smooth domain of \(\mathbb R^N,\) \(\lambda >0\) and \(g:[0,\infty)\to \mathbb R\) is smooth increasing, convex, \(g(0)>0\) and superlinear at \(+\infty.\) Some typical examples are \(g(u)=e^u\) and \(g(u)=(1+u)^p,\) \(p>1.\) The author explores up to some extend known results for this problem valid in other situations with a similar structure, with emphasis on the extremal solution and its properties. It is also considered the question of identifying conditions such that the extremal solution is singular. It is established that in the problems studied, there is a strong link between these conditions and Hardy-type inequalities.
For the entire collection see [Zbl 1179.35002].

Reviewer: Lubomira Softova (Aversa)

MSC:

35J61	Semilinear elliptic equations
35J25	Boundary value problems for second-order elliptic equations
35B20	Perturbations in context of PDEs
35J67	Boundary values of solutions to elliptic equations and elliptic systems
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

singular solutions; blow-up solutions; stability; perturbation of singular solutions