On the set of eigenvalues for some classes of coercive and noncoercive problems involving (2, p ( x ) )-Laplacian-like operators

• Published in: Electronic journal of qualitative theory of differential equations Vol. 2025; no. 31; pp. 1 - 21
• Main Author: Uța, Vasile
• Format: Journal Article
• Language: English
• Published: University of Szeged 2025
• Subjects: coercive and noncoercive case; continuous bounded or unbounded spectrum; double-phase differential operator; eigenvalue problem; variable exponent
• ISSN: 1417-3875; 1417-3875
• DOI: 10.14232/ejqtde.2025.1.31

We consider a class of double-phase nonlinear eigenvalue problems driven by a ( 2 , ϕ ) -Laplace-like operator: − Δ u − ε div ⁡ [ ϕ ( x , | ∇ u | ) ∇ u ] = λ ( u + ε ) in a domain Ω , subject to Dirichlet boundary conditions, where Ω is a bounded subset of R N with a smooth boundary. Here, ε > 0 , and the potential function ϕ exhibits p ( x ) -variable growth. We establish several results on the existence and concentration of eigenvalues for this problem, focusing on the influence of the growth behavior of the potential function ϕ , specifically through the interaction between the variable growth exponent p ( x ) and the constant growth exponent 2 . The proofs rely on variational arguments based on the Direct Method in the Calculus of Variations, Ekeland's variational principle, and energy estimates.