A System of p-Laplacian Equations on the Sierpinski Gasket

Abstract

In this paper we study a system of boundary value problems involving weak p-Laplacian on the Sierpiński gasket in R 2 . Parameters λ, γ, α, β are real and 1 < q < p < α + β. Functions a, b, h : S → R are suitably chosen. For p > 1 we show the existence of at least two nontrivial weak solutions to the system of equations for some (λ,for all (φ 1 , φ 2 ) ∈ dom 0 (E p ) × dom 0 (E p ), then we will call (u, v) to be a weak solution of (1.1). Differential equation on fractal domains has been of great interest for researcher for past few decades. We will go back in time line and give a brief review of literature survey. In [18,19,26,12], Laplacian is defined

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

“…Inspired by the definition of the energy form given by the inclusion (2.2) and by the ideas of Sahu and Priyadarshi, 44 we consider the following definition for weak solutions to (1.3).…”

“…However, (1.3) involves in addition the q-Laplacian, making particularly, that the functionals Φ and Θ 𝜑,𝛾 , introduced later by Equations 4.2 and (4.4), will not be homogeneous in the variable (u, v), which increases the difficulty of the study compared with that in Sahu and Priyadarshi. 44 Moreover, the assumption a, b ≥ 0 is no longer compulsory and the hypothesis (H3) of Sahu and Priyadarshi 44 is omitted in our paper. We note that problem (1.3) is a generalization of the following one with a single equation…”

“…Later et al 43 presented a unified approach to the method and gave some of its applications to nonlinear boundary value problems. Nevertheless, the fibering was one-dimensional in Sahu and Priyadarshi 44 where they treated the two-parametric system…”