In this paper, we study the following boundary value problem involving the weak \(p\)-Laplacian.$$ -M\bigl(\|u\|_{\mathcal{E}_{p}}^{p}\bigr)\Delta _{p} u = h(x,u) \quad \text{in}\ \mathcal{S}\setminus \mathcal{S}_{0}; \quad u = 0 \ \text{on}\ \mathcal{S}_{0}, $$where \(\mathcal{S}\) is the Sierpiński gasket in \(\mathbb{R}^{2}\), \(\mathcal{S}_{0}\) is its boundary, \(M: \mathbb{R}^{+} \to \mathbb{R}\) is defined by \(M(t) = at^{k} +b\), where \(a,b,k >0\) and \(h: \mathcal{S}\times \mathbb{R}\to \mathbb{R}\). We will show the existence of two nontrivial weak solutions to the above problem.

中文翻译：

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Sierpiński垫片上Kirchhoff型p $ p $ -Laplacian方程的多重解的存在

在本文中，我们研究以下涉及弱\（p \）- Laplacian的边值问题。$$ -M \ bigl（\ | u \ | _ {\ mathcal {E} _ {p}} ^ {p} \ bigr）\ Delta _ {p} u = h（x，u）\ quad \ text {在} \ \ mathcal {S} \ setminus \ mathcal {S} _ {0}中；\ quad u = 0 \ \ text {on} \ \ mathcal {S} _ {0}，$$其中\（\ mathcal {S} \）是\（\ mathbb {R} ^ {2}中的Sierpiński垫片\），\（\ mathcal {S} _ {0} \）是其边界，\（M：\ mathbb {R} ^ {+} \ to \ mathbb {R} \）由\（M（t ）= at ^ {k} + b \），其中\（a，b，k> 0 \）和\（h：\ mathcal {S} \ times \ mathbb {R} \至\ mathbb {R} \）。我们将展示上述问题的两个非平凡弱解的存在。