Anisotropic problems with unbalanced growth

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*). The proof combines variational methods, analytic tools, and monotonicity arguments.

Keywords

• nonlinear elliptic equation
• nonhomogeneous differential operator
• variable exponent
• weak solution
• mountain pass
• 35j60
• 35j65
• 35j70
• 58e05