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Zbl 0782.35053
Guo, Zongming
Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 18, No.10, 957-971 (1992). ISSN 0362-546X

The author proves the existence of solutions to the Dirichlet problem for $Au=\lambda f(u)$, where $A=-\text{div}(\vert Du\vert\sp{p-2})$ is the $p$-Laplacian and $\lambda$ is a positive parameter. The function $f$ vanishes at 0, and is either strictly increasing and $O(u\sp \mu)$ for some $\mu<p-1$, or has a single positive hump. Results for $p<2$ rely on a strong maximum principle, as in {\it P. Hess} [Commun. Partial Differ. Equations 6, 951-961 (1981; Zbl 0468.35073)] and the reviewer's paper [Houston J. Math. 16, No. 1, 139-149 (1990; Zbl 0717.47026)]. A few results for $p>2$, and on the necessity of the assumptions of $f$ are also included.
[S.Kichenassamy (Minneapolis)]
MSC 2000:
*35P30 Nonlinear eigenvalue problems for PD operators
35J70 Elliptic equations of degenerate type
47H11 Degree theory
35J65 (Nonlinear) BVP for (non)linear elliptic equations

Keywords: multiplicity results; degree theory; existence; Dirichlet problem; $p$- Laplacian; strong maximum principle

Citations: Zbl 0468.35073; Zbl 0717.47026

Cited in: Zbl 0834.34027

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