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Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. (English) Zbl 0822.35046

The authors study the number of solutions u of the problem

-εΔu+u=f(u),u>0inΩ,u=0onΩ,(P ε )

where ε>0 and f: + is subcritical and superlinear at 0 and at infinity. They show that there exists ε * >0 such that, if 0<εε * and all solutions of (P ε ) are non-degenerate, we have

u𝒦 t μ(u) =t𝒫 t (Ω)+t 2 [𝒫 t (Ω)-1]+t(1+t)𝒬(t),

where 𝒦 is the set of solutions of (P ε ), μ(u) is the Morse index of u, 𝒫 t (Ω) is the PoincarĂ© polynomial of Ω and 𝒬 is a suitable polynomial with non- negative integer coefficients. Actually, this is a consequence of a more general statement, where 𝒦 is required only to be discrete. Also an estimate of the cardinality of 𝒦 in terms of the Ljusternik- Schnirelman category of Ω is given.


MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
35J20Second order elliptic equations, variational methods
58E05Abstract critical point theory
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