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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.

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
Full Text: DOI
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