On the stationary solutions of generalized reaction diffusion equations with $p& q$-Laplacian

In this paper, we analyse a family of stationary nonlinear equations with $p& q$- Laplacian -Δₚ u -Δq u=λ c(x,u) which have a wide spectrum of applications in many areas of science. We introduce a new type of variational principles corresponding to this family of equations. Using this formalism, we exhibit intervals for the scalar parameter λ where there exist positive solutions of the considered problems. Furthermore, we prove, in another interval, the nonexistence of nontrivial solutions. These results are different from those of existence and nonexistence for stationary equations with single Laplacian.