Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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BIHARMONIC-KIRCHHOFF TYPE EQUATION INVOLVING CRITICAL SOBOLEV EXPONENT WITH SINGULAR TERM

  • Tahri, Kamel (High School of Management, Tlemcen Abou Bekr Belkaid University Faculty of Sciences Mathematics Department Rocade, Tlemcen Laboratory of Dynamic System and Applications) ;
  • Yazid, Fares (Department of Mathematics Laboratory of Pure and Applied Mathematics Amar Teledji University)
  • Received : 2020.04.30
  • Accepted : 2020.11.17
  • Published : 2021.04.30
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Abstract

Using variational methods, we show the existence of a unique weak solution of the following singular biharmonic problems of Kirchhoff type involving critical Sobolev exponent: $$(\mathcal{P}_{\lambda})\;\{\begin{array}{lll}{\Delta}^2u-(a{\int}_{\Omega}{\mid}{\nabla}u{\mid}^2dx+b){\Delta}u+cu=f(x){\mid}u{\mid}^{-{\gamma}}-{\lambda}{\mid}u{\mid}^{p-2}u&&\text{ in }{\Omega},\\{\Delta}u=u=0&&\text{ on }{\partial}{\Omega},\end{array}$$ where Ω is a smooth bounded domain of ℝn (n ≥ 5), ∆2 is the biharmonic operator, and ∇u denotes the spatial gradient of u and 0 < γ < 1, λ > 0, 0 < p ≤ 2# and a, b, c are three positive constants with a + b > 0 and f belongs to a given Lebesgue space.

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Acknowledgement

The authors would like to thank the anonymous referees for very helpful suggestions and comments which lead to the improvement of this paper.

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