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Some existence and multiplicity results for a class of quasilinear elliptic eigenvalue problems. (English) Zbl 0782.35053
The author proves the existence of solutions to the Dirichlet problem for $$Au=\lambda f(u)$$, where $$A=-\text{div}(| Du|^{p-2})$$ is the $$p$$-Laplacian and $$\lambda$$ is a positive parameter. The function $$f$$ vanishes at 0, and is either strictly increasing and $$O(u^ \mu)$$ for some $$\mu<p-1$$, or has a single positive hump. Results for $$p<2$$ rely on a strong maximum principle, as in 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.

##### MSC:
 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J70 Degenerate elliptic equations 47H11 Degree theory for nonlinear operators 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Citations:
Zbl 0468.35073; Zbl 0717.47026
Full Text:
##### References:
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