Infinitely Many Solutions for the Dirichlet Problem on the Sierpinski Gasket

Abstract

We study the nonlinear elliptic equation Δu(x) + a(x)u(x) = g(x)f(u(x)) on the Sierpinski gasket and with zero Dirichlet boundary condition. By extending a method introduced by Faraci and Kristály in the framework of Sobolev spaces to the case of function spaces on fractal domains, we establish the existence of infinitely many weak solutions.

Nonlinear problems on the Sierpiński gasket

“…In this section we briefly recall some basic facts on the Sierpiński gasket V and the functional space H 1 0 (V ) firstly introduced in [15] (see also [5,6,7,8,9,10,21]). …”

“…Let µ be the restriction to V of the normalized log N/ log 2-dimensional Hausdorff measure H d on R N −1 , so that µ(V ) = 1 (see, for instance, Breckner, Rȃdulescu and Varga [7] for more details). Finally, we also recall the following property of µ which will be useful in the sequel:…”

“…See, among others, the papers [5,6,7,8,9,10,13] and [15,17,18,19,21,28,29], as well as the references therein, where the authors obtained several existence and multiplicity results for problems on fractal domains under different growth assumptions on the data. The first and the third author were supported by the INdAM-GNAMPA Project 2016 Problemi variazionali su varietà riemanniane e gruppi di Carnot, by the DiSBeF Research Project 2015 Fenomeni non-locali: modelli e applicazioni, by the DiSPeA Research Project 2016 Implementazione e testing di modelli di fonti energetiche ambientali per reti di sensori senza fili autoalimentate and by the PRIN 2015 Research Project Variational methods, with applications to problems in mathematical physics and geometry.…”

Variational analysis for a nonlinear elliptic problem on the Sierpiński gasket

“…Finally, we recall that Breckner et al [7] proved the existence of infinitely many solutions of problem (S f,g a,λ ) under the key assumption, among others, that the non-linearity f is non-positive in a sequence of positive intervals. We point out that our results are mutually independent compared to those achieved in the above mentioned manuscript; see Remark 4.2.…”

Two Nontrivial Weak Solutions for the Dirichlet Problem on the Sierpiński Gasket

“…They have constructed and investigated Brownian motion on the Sierpiński gasket. In their standpoint, the Laplace operator 396 D. Stancu-Dumitru [2] has been formulated as the infinitesimal generator of the diffusion process. On the other hand, a direct and natural construction of a Laplacian on the Sierpiński gasket as a limit of difference quotients was given by Kigami [7], who later extended the method to the class of post critically finite fractals; for details see [8,9].…”