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
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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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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DOI: https://doi.org/10.1007/s41808-022-00165-w
Keywords
- Neumann problem
- Weak solution
- p(x)-Kirchhoff-Laplace
- Topological degree methods
- Variable exponent Sobolev spaces