Journal of Convex Analysis 27 (2020), No. 4, 1363--1374
Copyright Heldermann Verlag 2020
Multiplicity of Positive Solutions for an Anisotropic Problem via Sub-Supersolution Method and Mountain Pass Theorem
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
[Abstract-pdf]
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 \begin{equation*} \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) \mbox{ in } \Omega\mbox{,}\\[1mm] \ u>0\mbox{ in }\Omega, \quad u=0\mbox{ on } \partial\Omega\mbox{.} \end{cases} \end{equation*} We also prove the uniqueness of the solution for the linear anisotropic problem, a Comparison Principle for the anisotropic operator and a regularity result.
Keywords: Anisotropic operator, sub-supersolution method, Mountain Pass Theorem.
MSC: 35J60; 35J66.
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