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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
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