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

Abstract

In this paper, we establish the existence and multiplicity results of solutions for parametric quasi-linear systems of the gradient-type on the Sierpiński gasket is proved. Our technical approach is based on variational methods and critical points theory and on certain analytic and geometrical properties of the Sierpiński fractal. Indeed, using a consequence of the local minimum theorem due to Bonanno, the Palais-Smale condition cut off upper at r, and the Palais-Smale condition for the Euler functional we investigate the existence of one and two solutions for our problem under algebraic conditions on the nonlinear part. Moreover by applying a different three critical point theorem due to Bonanno and Marano we guarantee the existence of third solution for our problem.

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

“…We mention the work of Breckner and Varga 23 who treated a gradient-type system with one parameter by applying a critical point theorem due to Ricceri. 34 For Alrikabi et al, 35 it was a question of another type of one parametric system involving the Laplacian, using existence theorems due to Bonanno 36,37 and Bonanno and Marano. 38 The bibliography is also poor when speaking about looking for one or more nontrivial weak solutions to p-Laplacian systems on fractals.…”