Existence Results for Nonlinear Elliptic Equations on the Sierpi´nski Gasket via Monotone Operators

Abstract

In this work, we discuss the existence of weak solutions for the following class of nonlinear elliptic problem −∆w−λg(y)w + h(w) = f(y) in V \V0 w =0 onV0, (1) where V is the Sierpi´nski gasket in RN−1(N ≥ 2), V0 denotes its boundary (consisting of its N corners), ∆ is the Laplacian operator on V , λ ∈ R and f, g : V →R, h: R → R are the functions satisfying the following assumptions: (H1) f ∈ L2(Ω),g ∈ L∞(Ω), (H2) h : R → R be a Lipschitz continuous function with Lipschitz constant D and h(0) = 0, (H3) h satisfies (h(η) − h(η′))(η − η′) ≥ 0, for all η, η′ ∈ R. By utilizing monotone operator frameworks, we address the existence and qualitative properties of solutions to nonlinear elliptic equations on fractal domains.