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Mathematische Annalen

, Volume 348, Issue 1, pp 143–193 | Cite as

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

  • Juan Dávila
  • Isabel Flores
  • Ignacio Guerra
Article

Abstract

Let B be the unit ball in \({\mathbb{R}^N}\), N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to
$$\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B$$
where λ ≥ 0. For positive p we assume 5 ≤ N ≤ 12 and \({p > \frac{N+4}{N-4}}\), or N ≥ 13 and \({\frac{N+4}{N-4} < p < p_c}\), where p c is a constant depending on N. For negative p we assume 4 ≤ N ≤ 12 and p < p c , or N = 3 and \({p_{c}^{+} < p < p_c}\) , where \({p_{c}^{+}}\) is a constant. We show that there is a unique λ S > 0 such that if λλ S there exists a radial weakly singular solution. For λλ S there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as λλ S .

Mathematics Subject Classification (2000)

35J60 35J40 35B32 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Departamento de Ingenierí a Matemática and Centro de Modelamiento Matemático (UMI 2807, CNRS)Universidad de ChileSantiagoChile
  2. 2.Departamento de Matemática, Facultad de Ciencias Físicas y MatemáticasUniversidad de ConcepciónConcepciónChile
  3. 3.Departamento de Matematica y C.C., Facultad de CienciaUniversidad de Santiago de ChileSantiagoChile

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