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Lane-Emden equations and related topics on nonlinear elliptic and parabolic problems. (English) Zbl 0674.35047

New results in nonlinear partial differential equations, Semin. Bonn/FRG 1984, Aspects. Math. E10, 135-152 (1987).
[For the entire collection see Zbl 0606.00011.]
This is an expository paper on existence and nonexistence of the solution of the equation \[ (1)\quad \Delta u+u^ p=0,\quad u>0\quad in\quad \Omega;\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is bounded smooth domain of \({\mathbb{R}}^ n.\)
This equation arises in various branches of applied mathematics. For instance, in astrophysics it is known a Lane-Emden equation and u represents the density of stars.
Since only radial solutions on balls are of interest to astrophysicists, results on equation (1) are given and also the following general form of (1) is studied: \[ (2)\quad \Delta u+f(u,| x|)=0\quad in\quad \Omega;\quad u>0\quad in\quad \Omega \quad and\quad u=0\quad on\quad \partial \Omega, \] where f(u,\(| x|)\) behaves like \(| x| u^ p\) or more generally.
An E-solution of (2) is a continuous solution of (2) when \(\Omega\) is a ball, an F-solution of (2) is a classical solution of (2) when \(\Omega\) is an annulus. Finally an M-solution of (2) is a solution of \[ \Delta u+f(u,| x|)=0\quad in\quad \Omega \setminus \{0\},\quad u>0\quad in\quad \Omega \setminus \{0\},\quad u=0\quad on\quad \partial \Omega \quad and\quad u\to \infty \quad as\quad | x| \to 0, \] where \(\Omega\) is a ball.
E-, F-, M-solutions of (2) are discussed and conditions on p, n, \(\ell\) to have existence, nonexistence or only uniqueness of such solutions are mentioned.
Reviewer: S.Totaro

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
49K20 Optimality conditions for problems involving partial differential equations
85A15 Galactic and stellar structure
35L70 Second-order nonlinear hyperbolic equations

Citations:

Zbl 0606.00011