BOUNDARY VALUE PROBLEM INVOLVING THE p-LAPLACIAN ON THE SIERPIŃSKI GASKET

Abstract

In this paper, we study the following boundary value problem involving the weak [Formula: see text]-Laplacian. [Formula: see text] [Formula: see text] where [Formula: see text] is the Sierpiński gasket in [Formula: see text], [Formula: see text] is its boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are bounded nonnegative functions. We will show the existence of at least two nontrivial weak solutions to the above problem for a certain range of [Formula: see text] using the analysis of fibering maps on suitable subsets.

On the existence of solutions to a (p,q)‐Laplacian system on the Sierpiński gasket on ℝN−1

“…Lemma 32 There exists a constant $$$ {K}_s>0 $$$, such that $$$$ {\left\Vert u\right\Vert}_{\infty}\le {K}_s{\left\Vert u\right\Vert}_s,\kern2em \forall u\in {D}^s. $$$$ …”

“…As mentioned in previous research, 31,32 we do not know whether  s (u + tv) is differentiable or not. But the convexity of A s implies that  s is also a convex function.…”

“…As mentioned in previous research, 31,32 we do not know whether $$$ {\mathcal{E}}_s\left(u+ tv\right) $$$ is differentiable or not. But the convexity of $$$ {A}_s $$$ implies that $$$ {\mathcal{E}}_s $$$ is also a convex function.…”

“…Then, Strichartz and Wong introduced the $$$ p $$$‐Laplacian operator on this family of PCF fractal domains 31 . This allowed some researchers to study elliptic equations including this kind of operators on these domains (see, e.g., previous study 32,33 ).…”

A System of p-Laplacian Equations on the Sierpinski Gasket

“…Priyadarshi and Sahu [23] studied the problem −∆ p u = λa(x)|u| q−1 u + b(x)|u| l−1 u in S \ S 0 , u = 0 on S 0 , and have shown the existence of two solutions under the assumptions p > 1, 0 < q < p − 1 < l, a, b : S → R are bounded nonnegative functions for a small range of λ. Many authors have tried to address problems on fractal domains (see, for example, [12,5,21]).…”