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

The authors study quasilinear problems

-div(|Du| p-2 Du)=λf(u)onΩ,u=0onΩ,(*)

where Ω is a bounded domain in N with a smooth boundary Ω, p>1, λ>0, f a real function, f(u)>0 for u>0. Under certain further assumptions (mainly f is nondecreasing, lim s0 + inff(s)/s p-1 >0, (f(s)/s p-1 ) ' <0), the existence and uniqueness of a positive solution of (*) for λ large enough is shown. The proof is based on a generalization of the Serrin’s sweeping principle. If Ω is an annulus then the solution is radially symmetric.

MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35J25Second order elliptic equations, boundary value problems
35A05General existence and uniqueness theorems (PDE) (MSC2000)