Anisotropic problems with unbalanced growth

• Main Authors: Alsaedi Ahmed, Ahmad Bashir
• Format: Article
• Language: English
• Published: De Gruyter 2020-03-01
• Series: Advances in Nonlinear Analysis
• Subjects: nonlinear elliptic equation; nonhomogeneous differential operator; variable exponent; weak solution; mountain pass; 35j60; 35j65; 35j70; 58e05
• Online Access: https://doi.org/10.1515/anona-2020-0063

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.