Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

Leonelo Iturriaga; Eugenio Massa

doi:10.3934/dcds.2018166

Article Contents

2018, Volume 38, Issue 8: 3831-3850. Doi: 10.3934/dcds.2018166

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Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

Leonelo Iturriaga^1, and
Eugenio Massa^2, ,

1.
Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile
2.
Departamento de Matemática, Instituto de Ciêencias Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

* Corresponding author: E. Massa
* Corresponding author: E. Massa

Received: August 2017

Revised: April 2018

Published: May 2018

The first author gratefully acknowledges financial support from Fondecyt grants 1161635, 1171532 and 1171691
The author E. Massa was supported by: grant #2014/25398-0, São Paulo Research Foundation (FAPESP) and grants #308354/2014-1, #303447/2017-6, CNPq/Brazil

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In this paper we consider the equation $(-Δ)^k\, u = λ f(x, u)+μ g(x, u)$ with Navier boundary conditions, in a bounded and smooth domain. The main interest is when the nonlinearity is nonnegative but admits a zero and $f, g$ are, respectively, identically zero above and below the zero. We prove the existence of multiple positive solutions when the parameters lie in a region of the form $λ>\overline λ$ and $0 < μ< \overlineμ(λ)$, then we provide further conditions under which, respectively, the bound $\overlineμ(λ)$ is either necessary, or can be removed.

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Mathematics Subject Classification: Primary: 35J40; Secondary: 35J91.

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