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

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