Abstract

We use the sub-supersolution method and the Mountain Pass Theorem in order to show existence and multiplicity of solution for an anisotropic problem given by { −[∑i=1N∂∂xi(∣∂u∂xi∣pi−2∂u∂xi) ]=a(x)u+h(x,u) in Ω, u>0 in Ω,u=0 on ∂Ω.\begin{cases} \ -\Big[\displaystyle\sum^{N}_{i=1}\frac{\partial}{\partial x_{i}} \Big( \Big\vert \frac{\partial u}{\partial x_{i}}\Big\vert^{pi-2} \frac{\partial u}{\partial x_{i}}\Big )\ \Big]=a(x)u+ h(x,u) \text{ in } \Omega\text{,}\\[1mm] \ u>0\text{ in }\Omega, \quad u=0\text{ on } \partial\Omega\text{.} \end{cases} We also prove the uniqueness of the solution for the linear anisotropic problem, a Comparison Principle for the anisotropic operator and a regularity result.

Author information

Contact details are reproduced from the original publication and may be historical.

Gelson C. G. dos Santos

Faculdade de Matemática, Universidade Federal do Pará, 66075-110 Belém-Pa, Brazil

cgelson@ymail.com

Giovany Figueiredo

Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília-DF, Brazil

giovany@unb.br

Julio R. S. Silva

Universidade Federal do Pará, Campus Universitário, 68.400-000 Cametá, Brazil

julioroberto@ufpa.br

Keywords

  • Anisotropic operator
  • sub-supersolution method
  • Mountain Pass Theorem

Mathematics Subject Classification

35J60; 35J66

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Suggested citation

G. C. G. dos Santos, G. Figueiredo, J. R. S. Silva. “Multiplicity of Positive Solutions for an Anisotropic Problem via Sub-Supersolution Method and Mountain Pass Theorem.” Journal of Convex Analysis 27 (2020), No. 4, 1363–1374. https://doi.org/10.68381/jca27072

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Published by Heldermann Verlag, 2020. Rights now held by Banach Press LLC.