Multiple solutions of inhomogeneous H-systems with zero Dirichlet boundary conditions

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Abstract

In this paper, we study the existence of multiple solutions to the Dirichlet problem of the inhomogeneous H-system:Δu=2HuxΛuy+finΩ,u|∂ Ω=0,where Ω⊂R2 is a bounded smooth domain, H > 0 a constant, and f ∈ H-1(Ω;R3) is a given function. By Ekeland's variational principle and the Mountain Pass Theorem, we prove that, for f≢0 satisfying some assumptions, the problem has at least two solutions in H01(Ω;R3). This is not the case if f≡0 and Ω is simply-connected [16].

Original languageEnglish
Pages (from-to)239-259
Number of pages21
JournalNonlinear Analysis, Theory, Methods and Applications
Volume52
Issue number1
DOIs
Publication statusPublished - Jan 1 2003
Externally publishedYes

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Ekeland's Variational Principle
Mountain Pass Theorem
Multiple Solutions
Dirichlet Problem
Dirichlet Boundary Conditions
Boundary conditions
Zero

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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abstract = "In this paper, we study the existence of multiple solutions to the Dirichlet problem of the inhomogeneous H-system:Δu=2HuxΛuy+finΩ,u|∂ Ω=0,where Ω⊂R2 is a bounded smooth domain, H > 0 a constant, and f ∈ H-1(Ω;R3) is a given function. By Ekeland's variational principle and the Mountain Pass Theorem, we prove that, for f≢0 satisfying some assumptions, the problem has at least two solutions in H01(Ω;R3). This is not the case if f≡0 and Ω is simply-connected [16].",
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AB - In this paper, we study the existence of multiple solutions to the Dirichlet problem of the inhomogeneous H-system:Δu=2HuxΛuy+finΩ,u|∂ Ω=0,where Ω⊂R2 is a bounded smooth domain, H > 0 a constant, and f ∈ H-1(Ω;R3) is a given function. By Ekeland's variational principle and the Mountain Pass Theorem, we prove that, for f≢0 satisfying some assumptions, the problem has at least two solutions in H01(Ω;R3). This is not the case if f≡0 and Ω is simply-connected [16].

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