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

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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 λ = λ S there exists a radial weakly singular solution. For λ = λ S there exist infinitely many regular radial solutions and the number of radial regular solutions goes to infinity as λ → λ S .

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Authors and Affiliations

  1. Departamento de Ingenierí a Matemática and Centro de Modelamiento Matemático (UMI 2807, CNRS), Universidad de Chile, Casilla 170/3 Correo 3, Santiago, Chile

    Juan Dávila

  2. Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160 C, Concepción, Chile

    Isabel Flores

  3. Departamento de Matematica y C.C., Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

    Ignacio Guerra

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  1. Juan Dávila
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  2. Isabel Flores
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Correspondence to Juan Dávila.

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Dávila, J., Flores, I. & Guerra, I. Multiplicity of solutions for a fourth order equation with power-type nonlinearity. Math. Ann. 348, 143–193 (2010). https://doi.org/10.1007/s00208-009-0476-8

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  • DOI: https://doi.org/10.1007/s00208-009-0476-8

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