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A Variational Approach to a Kirchhoff-type Problem Involving Two Parameters

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Abstract

In this paper, the authors establish the existence of at least three weak solutions for the Kirchhoff-type problem

$$\left\{\begin{array}{ll}-K \left( \int_{\Omega}| \nabla u(x)|^{2}dx \right) \Delta u(x)= \lambda f(x,u)+\mu g(x,u),\quad {\rm in}\; \Omega,\\u=0, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad {\rm on}\; \partial \Omega, \end{array} \right.$$

under appropriate hypotheses. The proofs are based on variational methods.

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, 37403, USA

    John R. Graef & Lingju Kong

  2. Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah, 67149, Iran

    Shapour Heidarkhani

  3. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5746, Tehran, Iran

    Shapour Heidarkhani

Authors
  1. John R. Graef
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  2. Shapour Heidarkhani
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  3. Lingju Kong
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Corresponding author

Correspondence to John R. Graef.

Additional information

S. Heidarkhani’s research supported in part by IPM Grant No. 89350020.

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Graef, J.R., Heidarkhani, S. & Kong, L. A Variational Approach to a Kirchhoff-type Problem Involving Two Parameters. Results. Math. 63, 877–889 (2013). https://doi.org/10.1007/s00025-012-0238-x

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  • DOI: https://doi.org/10.1007/s00025-012-0238-x

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