Research of the Fučik spectrum for the (p,q)-Laplacian operator by min-max theory

Abstract

The object of research is the Fučik spectrum for the (p,q)-Laplacian operator. In the present paper, we are going to introduce the notion of the Fučik spectrum for a non-linear, non-homogeneous operator, which is the (p,q)-Laplacian operator through the study of the following eigenvalue boundary problem: {–∆pu–∆qu=λ(u+)p–1–μ(u–)q–1 in Ω, u=0 on ∂Ω, where Ω⊂RN, N≥1 is a bounded open subset with smooth boundary and λ and μ are two real parameters. In order to establish and show the existence of non-trivial solutions for the problem described above, we will search the weak solution of the energy functional associated to our problem by combining two essentials theorems of the Min-Max theory which are the Ljusternick-Schnirelmann (L-S) approach and the Col theorem. In addition to that, we are going to use the Ljusternick-Schnirelman theorem to show that our problem possesses a critical value ck in a suitable manifold that we will define later in the present manuscript. Following to that we will verify the Col geometry by using the critical point associated to the critical value ck and by applying the Col theorem, we will find a new critical value cn. After that, by employing the critical value cn we will demonstrate the existence of the family of curves which generate the set of Fučik spectrum of the (p,q)-Laplacian operator. To complete our research about the structure of the set of the Fučik spectrum of the (p,q)-Laplacian operator we will give the most important properties of the family of curves which are the continuity and the decrease. We have chosen to put our interest on the study of the Fučik spectrum because it’s determination is as important in mathematics as it is in many other fields (physics, plasma-physics, reaction-diffusion equation etc.). We can take as an example it’s use in the field of waves and vibrations where the starting point of the wave or the vibration is influenced by the structure and characteristics of the family of curves which constitute the Fučik spectrum of the (p,q)-Laplacian operator.