Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large. (English) Zbl 0799.35081

The authors study quasilinear problems \[ -\text{div} (| Du|^{p- 2} Du) =\lambda f(u) \quad \text{on } \Omega, \qquad u=0 \quad \text{on }\partial \Omega,\tag{*} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ N\) 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, \(\lim_{s\to 0_ +} \inf f(s)/ s^{p- 1}>0\), \((f(s)/ s^{p-1})'<0\)), 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.
Reviewer: M.Kučera (Praha)


35J65 Nonlinear boundary value problems for linear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI


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