Concentrating solutions for a class of indefinite Schrödinger-Poisson systems with doubly critical growth - Shen, Liejun; Squassina, Marco (Topological Methods in Nonlinear Analysis, 2025)
We study the following Schrödinger-Poisson system involving critical nonlocal term with indefinite steep potential well cases -Δ u+(λ V(x)-μ)u-φ |u|^3u= f(u), & x^3, -Δ φ= |u|^5, & x^3, cases % where λ> 0 is a parameter, VC(^3,^+) admits a potential well Ω riangleqintV^-1(0), and μ> μ_1 is a constant such that the operator L_λ riangleq -Δ + λ V-μ is non-degenerate when λ is large enough with {μ_j}_j=1^∞ denoting the Dirichlet eigenvalues of (-Δ,H_0^1(Ω)). If f satisfies some suitable assumptions involving critical growth, with the help of a linking-type result involving the modified Pankov-Nehari manifold procedure, we establish the existence and concentrating behavior of positive solutions for the given system using variational methods.