Nontrivial Solution for Indefinite Quasilinear Schrödinger-Poisson Systems

In this paper, we investigate Quasilinear Schrödinger-Poisson systems: { −Δu+V( x )u+ϕu=f( x,u ), x∈ ℝ 3 , −Δϕ− ε 4 Δ 4 ϕ= u 2 , x∈ ℝ 3 , where the potential V is indefinite, leading the Schrödinger operator −Δ+V exhibit a finite-dimensional negative space. The presence of such a potential causes the spectrum of the corresponding Schrödinger operator −Δ+V to include a negative part, thereby generating a finite-dimensional negative within the variational framework. As a result, the energy functional exhibits a saddle-point structure at the origin, which breaks the classical mountain pass geometry and prevents the direct application of the mountain pass lemma. To overcome this difficulty, we instead exploit the local linking properties of the functional and employ Morse theory, ultimately proving the existence of nontrivial solutions for the system.