Anisotropic problems with unbalanced growth

Abstract

Abstract The main purpose of this paper is to study a general class of (p, q)-type eigenvalues problems with lack of compactness. The reaction is a convex-concave nonlinearity described by power-type terms. Our main result establishes a complete description of all situations that can occur. We prove the existence of a critical positive value λ * such that the following properties hold: (i) the problem does not have any entire solution in the case of low perturbations (that is, if 0 < λ < λ *); (ii) there is at least one solution if λ = λ *; and (iii) the problem has at least two entire solutions in the case of high perturbations (that is, if λ > λ *). The proof combines variational methods, analytic tools, and monotonicity arguments.