Existence and multiplicity of positive solutions for the p-Laplacian with nonlocal coefficient.

• Autor(a) principal: Bueno, H.
• Data de Publicação: 2008
• Outros Autores: Ercole, Grey, Ferreira, Wenderson Marques, Santos, Antônio Zumpano Pereira
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Institucional da UFOP
• Texto Completo: http://www.repositorio.ufop.br/handle/123456789/4597; https://doi.org/10.1016/j.jmaa.2008.01.001
• topic: Laplacian; Nonlocal coefficient; Existence and multiplicity of positive solutions

We consider the Dirichlet problem with nonlocal coefficient given by −a(Ω|u|q dx)_pu = w(x)f (u) in a bounded, smooth domain Ω ⊂ Rn (n _ 2), where _p is the p-Laplacian, w is a weight function and the nonlinearity f (u) satisfies certain local bounds. In contrast with the hypotheses usually made, no asymptotic behavior is assumed on f . We assume that the nonlocal coefficient a(_Ω|u|q dx) (q _ 1) is defined by a continuous and nondecreasing function a : [0,∞)→[0,∞) satisfying a(t) > 0 for t > 0 and a(0) _ 0. A positive solution is obtained by applying the Schauder Fixed Point Theorem. The case a(t) = tγ/q (0 < γ < p − 1) will be considered as an example where asymptotic conditions on the nonlinearity provide the existence of a sequence of positive solutions for the problem with arbitrarily large sup norm.