Ukrainian Mathematical Journal

, Volume 71, Issue 4, pp 651–662 | Cite as

Existence Results for a Class of Kirchhoff-Type Systems with Combined Nonlinear Effects

  • G. A. AfrouziEmail author
  • S. Shakeri
  • H. Zahmatkesh
We study the existence of positive solutions for a nonlinear system
$$ {\displaystyle \begin{array}{l}-{M}_1\left(\underset{\varOmega }{\int }{\left|{\nabla}_u\right|}^p dx\right)\operatorname{div}\left({\left|x\right|}^{- ap}{\left|{\nabla}_u\right|}^{p-2}{\nabla}_u\right)=\lambda {\left|x\right|}^{-\left(a+1\right)p+{c}_1}f\left(u,\upsilon \right),\kern1em x\in \varOmega, \\ {}-{M}_2\left(\underset{\varOmega }{\int }{\left|{\nabla}_{\upsilon}\right|}^q dx\right)\operatorname{div}\left({\left|x\right|}^{- bq}{\left|{\nabla}_{\upsilon}\right|}^{q-2}{\nabla}_{\upsilon}\right)=\lambda {\left|x\right|}^{-\left(b+1\right)q+{c}_2}g\left(u,\upsilon \right),\kern1em x\in \varOmega, \\ {}u=\upsilon =0,\kern1em x\in \vartheta \varOmega, \end{array}} $$

where Ω is a bounded smooth domain in ℝN with \( 0\in \varOmega, 1\kern0.33em <\kern0.33em p,q\kern0.33em <N,0\le a<\frac{N-p}{p},0\le b<\frac{N-q}{q}, \) and c1, c2, and λ are positive parameters. Here, M1, M2, f, and g satisfy certain conditions. We use the method of sub- and supersolutions to establish our results.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MazandaranBabolsarIran
  2. 2.Department of Mathematics, Ayatollah Amoli BranchIslamic Azad UniversityAmolIran

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