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

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Ukrainian Mathematical Journal Aims and scope

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. Department of Mathematics, University of Mazandaran, Babolsar, Iran

    G. A. Afrouzi & H. Zahmatkesh

  2. Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran

    S. Shakeri

Authors
  1. G. A. Afrouzi
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  2. S. Shakeri
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  3. H. Zahmatkesh
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Correspondence to G. A. Afrouzi.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 4, pp. 571–580, April, 2019.

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Afrouzi, G.A., Shakeri, S. & Zahmatkesh, H. Existence Results for a Class of Kirchhoff-Type Systems with Combined Nonlinear Effects. Ukr Math J 71, 651–662 (2019). https://doi.org/10.1007/s11253-019-01668-x

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  • DOI: https://doi.org/10.1007/s11253-019-01668-x

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