Let us consider the quasilinear problem (Pε)⎧⎩⎨⎪⎪−εpΔpu+up−1=f(u)u&gt;0u=0in Ω,in Ω,on ∂Ω, where Ω is a bounded domain in RN with smooth boundary, N≥2, 1&lt;2, ε&gt;0 is a parameter and f:R→R is a continuous function with f(0)=0, having a subcritical growth. We prove that there exists ε∗&gt;0 such that, for every ε∈(0,ε∗), (Pε) has at least 2P1(Ω)−1 solutions, possibly counted with their multiplicities, where Pt(Ω) is the Poincaré polynomial of Ω. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on Ω, approximating (Pε).

### Multiple positive solutions for a p-Laplace Benci–Cerami type problem (1<2), via Morse theory

#### Abstract

Let us consider the quasilinear problem (Pε)⎧⎩⎨⎪⎪−εpΔpu+up−1=f(u)u>0u=0in Ω,in Ω,on ∂Ω, where Ω is a bounded domain in RN with smooth boundary, N≥2, 1<2, ε>0 is a parameter and f:R→R is a continuous function with f(0)=0, having a subcritical growth. We prove that there exists ε∗>0 such that, for every ε∈(0,ε∗), (Pε) has at least 2P1(Ω)−1 solutions, possibly counted with their multiplicities, where Pt(Ω) is the Poincaré polynomial of Ω. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on Ω, approximating (Pε).
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11589/226697`
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