zbMATH — the first resource for mathematics

On the elliptic problem \(\Delta u-| \nabla u| ^ q+\lambda u^ p=0\). (English) Zbl 0699.35102
Nonlinear diffusion equations and their equilibrium states, I, Proc. Microprogram, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 12, 237-243 (1988).
[For the entire collection see Zbl 0643.00015.]
The authors study regular solutions of the problem \[ \Delta u-| \nabla u|^ q+\lambda u^ p=0,\quad x\in \Omega;\quad u>0,\quad x\in \Omega;\quad u=0,\quad x\in \partial \Omega. \] For various ranges of p and q the authors prove existence and uniqueness theorems. Interestingly, some of the results depend on the size of the support of the solution.
Reviewer: St.G.Krantz

35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)