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Existence and uniqueness results for a class of p(x)-Kirchhoff-type problems with convection term and Neumann boundary data

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Abstract

We establish an existence and uniqueness results for a homogeneous Neumann boundary value problem involving the p(x)-Kirchhoff-Laplace operator of the following form

$$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \displaystyle -M\Big ( \int _{{\Omega} }\frac{1}{p(x)}(\vert \nabla u\vert ^{p(x)}+\vert u\vert ^{p(x)})\,dx\Big ) \Big (\mathrm{{div}}(\vert \nabla u\vert ^{p(x)-2}\nabla u)-\vert u\vert ^{p(x)-2}u\Big )\\ =f(x, u, \nabla u) & \mathrm {i}\mathrm {n}\ ,\\ \vert \nabla u\vert ^{p(x)-2}\frac{\partial u}{\partial \eta }=0 & \mathrm {o}\mathrm {n}\ \partial {{\Omega}} . \end{array}\right. \end{aligned}$$

where \({{\Omega}}\) is a smooth bounded domain in \(\mathbb {R}^{N}\), \(\frac{\partial u}{\partial \eta }\) is the exterior normal derivative, \(p(x)\in C_{+}({\overline{{\Omega} }})\) with \(p(x)\ge 2\). By means of a topological degree of Berkovits for a class of demicontinuous operators of generalized \((S_{+})\) type and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on f and M, we obtain a results on the existence and uniqueness of weak solution to the considered problem.

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Acknowledgements

The authors would like to thank the referees and the editor for their careful reading and useful comments and suggestions that substantially helped improving the quality of the paper.

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  1. Applied Mathematics and Scientific Computing Laboratory, Faculty of Sciences and Techniques, Sultan Moulay Slimane University, Beni Mellal, Morocco

    Chakir Allalou, Mohamed El Ouaarabi & Said Melliani

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  1. Chakir Allalou
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  2. Mohamed El Ouaarabi
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  3. Said Melliani
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Correspondence to Mohamed El Ouaarabi.

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Allalou, C., El Ouaarabi, M. & Melliani, S. Existence and uniqueness results for a class of p(x)-Kirchhoff-type problems with convection term and Neumann boundary data. J Elliptic Parabol Equ 8, 617–633 (2022). https://doi.org/10.1007/s41808-022-00165-w

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