Theoretical analysis demonstrates ground state solutions for discrete Schrödinger equations on lattice graphs, highlighting existence criteria beyond strict monotonicity assumptions.
This paper studies the nonlinear discrete Schrödinger equation −Δu+(V(x)−ρ|x|2+1)u=f(x,u) on the lattice graph ZN with N≥3 and ρ≥0, where V is a bounded periodic potential and 0 lies in a spectral gap of the operator −Δ+V. By constructing a minimizing Cerami sequence outside the Nehari–Pankov manifold via a diagonal argument, we establish the existence of ground state solutions and investigate their asymptotic behavior as ρ→0+. This paper introduces a new method to handle the lack of superquadratic growth and the limitations of strict monotonicity assumption, and we improve the existing results.
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Chen et al. (2026) studied this question.
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