×

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)

MSC:

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)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1112/plms/s3-53.3.429 · Zbl 0572.35040
[2] DOI: 10.1016/0362-546X(84)90080-4 · Zbl 0526.35032
[3] Boccardo, Comment. Math. Univ. Carolinae 30 pp 411– (1989)
[4] DOI: 10.1007/BF01455933 · Zbl 0576.35044
[5] Sakaguchi, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14 pp 403– (1987)
[6] DOI: 10.1016/0022-0396(89)90093-4 · Zbl 0708.34019
[7] DOI: 10.1007/BF01221125 · Zbl 0425.35020
[8] DOI: 10.1088/0951-7715/3/3/008 · Zbl 0719.35027
[9] DOI: 10.1016/0362-546X(92)90132-X · Zbl 0782.35053
[10] DOI: 10.1080/00036819208840139 · Zbl 0788.35049
[11] DOI: 10.1016/0022-0396(88)90068-X · Zbl 0661.35029
[12] DOI: 10.1080/03605309908820717 · Zbl 0715.35029
[13] Gilbarg, Elliptic Partial Differential Equations of Second Order (1977)
[14] DOI: 10.1016/0362-546X(91)90229-T · Zbl 0731.35039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.