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Existence of weak solutions to a p-Laplacian system on the Sierpiński gasket on \({\mathbb {R}}^2\)

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Abstract

We study the existence of weak solutions for the following boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _pu =\displaystyle \frac{\alpha }{ \alpha +\beta }a(x)|u|^{\alpha -2}u |v|^\beta & \quad {\mathrm{in}} \,\, {\mathcal {S}} {\setminus } {\mathcal {S}}_0, \\ -\Delta _p v =\displaystyle \frac{\beta }{\alpha +\beta }a(x) |u|^\alpha |v|^{\beta -2} v &\quad {\mathrm{in}} \,\, {\mathcal {S}} {\setminus } {\mathcal {S}}_0, \\ v= u =0 &\quad {\mathrm{in}} \,\,{\mathcal {S}}_0. \end{array}\right. \end{aligned}$$

where \({\mathcal {S}}\) is the Sirpiński gasket on \({\mathbb {R}}^2\), \({\mathcal {S}}_0\) is its boundary and \(\Delta _p\) is the weak p-Laplacian operator on fractal domain, under some conditions on the function a and the reals \(p, \alpha\) and \(\beta\).

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  1. Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, 3000, Sfax, Tunisia

    Chouhaïd Souissi

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Souissi, C. Existence of weak solutions to a p-Laplacian system on the Sierpiński gasket on \({\mathbb {R}}^2\). Bol. Soc. Mat. Mex. 29, 7 (2023). https://doi.org/10.1007/s40590-022-00475-y

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