Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket

Abstract

Abstract.Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinear term, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem on the Sierpiński gasket is proved. Our approach is based on variational methods and on some analytic and geometrical properties of the Sierpiński fractal. The abstract results are illustrated by explicit examples.

Existence results for nonlinear elliptic problems on fractal domains

“…We now state a useful property of the space H 1 0 (V ) which shows, together with the facts that (H 1 0 (V ), · ) is a Hilbert space and (4). Thus h • u ∈ H 1 0 (V ) and h • u ≤ L u .…”

Nonlinear problems on the Sierpiński gasket

“…We refer the interested reader to [3,4,21,26,27] and references therein for some applications of Ricceri's variational principle and to [20] for related topics on the variational methods used in this paper (see also the classical reference [11]). …”

Existence and multiplicity of weak solutions for gradient-type systems wıth oscillatory nonlinearities on the sierpinski gasket

“…where a : V → R, f : R → R and g : V → R are continuous functions with appropriate properties. In [13,14] authors studied the nonlinear problem ∆u + a(x)u = λg(x)f (u) in V /V 0 , u = 0 on V 0 , where V is the Sierpiński gasket, V 0 is its intrinsic boundary, △ denotes the weak Laplace operator, and λ is a positive real parameter, and f has an oscillatory behaviour either near the origin or at infinity, in [13] they established the existence of infinitely many solutions but in [14] they studied the existence of sequence of weak solutions. In [23], authors analysed the problem…”