Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent

In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power-like variable reaction term. One of the main features of our work is that the differential operator involves partial derivatives with different variable exponents, so that the functional-analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena related to the existence of a continuous spectrum in several distinct situations.