On the set of eigenvalues for some classes of coercive and noncoercive problems involving (2, p ( x ) )-Laplacian-like operators - Uța, Vasile (Electronic journal of qualitative theory of differential equations, 2025)
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.