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Existence of Multiple Solutions of a Kirchhoff Type p $p$ -Laplacian Equation on the Sierpiński Gasket
Acta Applicandae Mathematicae ( IF 1.2 ) Pub Date : 2019-07-29 , DOI: 10.1007/s10440-019-00283-z
Abhilash Sahu , Amit Priyadarshi

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:

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\(\ mathcal {S} \)\(\ mathbb {R} ^ {2}中的Sierpiński垫片\)\(\ mathcal {S} _ {0} \)是其边界,\(M:\ mathbb {R} ^ {+} \ to \ mathbb {R} \)\(M(t )= at ^ {k} + b \),其中\(a,b,k> 0 \)\(h:\ mathcal {S} \ times \ mathbb {R} \至\ mathbb {R} \)。我们将展示上述问题的两个非平凡弱解的存在。
更新日期:2019-07-29
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