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Existence and multiplicity results for a semilinear elliptic eigenvalue problem. (English) Zbl 0662.35045
The authors prove a number of results for the equation \[ -\Delta u=\lambda f(u)\quad in\quad \Omega,\quad u=0\quad on\quad \partial \Omega. \] They consider conditions for the existence of positive solutions when f changes sign, and existence and uniqueness of positive solutions for large \(\lambda\).
Reviewer’s remark: There is closely related work of the reviewer [Proc. Lond. Math. Soc., III. Ser. 53, 429-452 (1986; Zbl 0572.35040)] the reviewer and K. Schmitt [Proc. Am. Math. Soc. 101, 445-452 (1987)] and the second author [Proc. R. Soc. Edinb., Sect. A 108, 357-370 (1988)]. Note that most of Theorem 3 can be proved for any domain by using work of the author [loc. cit.] and that the techniques of this paper could be used to improve Theorem 2 of the work of the author [loc. cit.] to allow the nonlinearity to have much more general dependence on x.
Reviewer: E.N.Dancer

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J25 Boundary value problems for second-order elliptic equations
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