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Zbl 0662.35045
Clément, Philippe; Sweers, Guido
Existence and multiplicity results for a semilinear elliptic eigenvalue problem. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No.1, 97-121 (1987). ISSN 0391-173X

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 $\lambda$. \par 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 {\it 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.
[E.N.Dancer]

MSC 2000:: *35J65 (Nonlinear) BVP for (non)linear elliptic equations
35P30 Nonlinear eigenvalue problems for PD operators
35A05 General existence and uniqueness theorems (PDE)
35J25 Second order elliptic equations, boundary value problems

Keywords: multiplicity; semilinear; asymptotic behaviour; existence; positive solutions; uniqueness

Citations: Zbl 0572.35040

Cited in: Zbl 0882.35052 Zbl 0812.35008

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