Existence results for a class of p(x)-Kirchhoff problems

Mostafa Allaoui

doi:10.1556/012.2017.54.3.1369

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Studia Scientiarum Mathematicarum Hungarica

Volume/Issue:: Volume 54: Issue 3

Existence results for a class of p(x)-Kirchhoff problems

Author:

Mostafa Allaoui

Mostafa Allaoui University Mohamed I, FSTH, Department of Mathematics, Oujda, Morocco

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Pages:: 316–331

Online Publication Date:: Sep 2017

Publication Date:: 01 Sep 2017

Article Category:: Journal Article

DOI:: https://doi.org/10.1556/012.2017.54.3.1369

Keywords:: nonlocal problems; nonlinear elliptic operator

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This paper is concerned with the existence of solutions to a class of p(x)-Kirchhofftype equations with Robin boundary data as follows: $- M (\int_{Ω} \frac{1}{p (x)} {| \nabla_{u} |}^{p (x)} d x + \int_{\partial Ω} \frac{β (x)}{p (x)} {| \nabla_{u} |}^{p (x)} d σ) d i v ({| \nabla_{u} |}^{p (x) - 2} \nabla_{u}) = f (x, u)$ in Ω, ${| \nabla_{u} |}^{p (x) - 2} \frac{\partial u}{\partial v} + β (x) {| u |}^{p (x) - 2} u = 0$ on ∂Ω, where β ∈ L^∞ (∂Ω) and f : Ω × $ℝ$ → $ℝ$ satisfies the Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

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Studia Scientiarum Mathematicarum Hungarica

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Studia Scientiarum Mathematicarum Hungarica
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