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Zbl 0799.35081
Guo, Zongming; Webb, J.R.L.
Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large.
(English)
[J] Proc. R. Soc. Edinb., Sect. A 124, No.1, 189-198 (1994). ISSN 0308-2105; ISSN 1473-7124/e

The authors study quasilinear problems $$-\text{div} (\vert Du\vert\sp{p- 2} Du) =\lambda f(u) \quad \text{on } \Omega, \qquad u=0 \quad \text{on }\partial \Omega,\tag*$$ where $\Omega$ is a bounded domain in $\bbfR\sp N$ with a smooth boundary $\partial\Omega$, $p>1$, $\lambda>0$, $f$ a real function, $f(u)>0$ for $u>0$. Under certain further assumptions (mainly $f$ is nondecreasing, $\lim\sb{s\to 0\sb +} \inf f(s)/ s\sp{p- 1}>0$, $(f(s)/ s\sp{p-1})'<0$), the existence and uniqueness of a positive solution of (*) for $\lambda$ large enough is shown. The proof is based on a generalization of the Serrin's sweeping principle. If $\Omega$ is an annulus then the solution is radially symmetric.
[M.Kučera (Praha)]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J25 Second order elliptic equations, boundary value problems
35A05 General existence and uniqueness theorems (PDE)

Keywords: quasilinear elliptic equation; positive solution

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