On the Schrödinger--Bopp--Podolsky system with indefinite potential: ground states, multiplicity and exponential decay

2026/07/17 by Ting Xiao, Fan Wang, Li-Feng Yin
#math.AP

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Abstract

In this paper, we study the Schrödinger--Bopp--Podolsky system \begincases -Δu + V(x)u + ϕu = f(x,u), in ℝ³, -Δϕ+ a² Δ² ϕ= 4πu², in ℝ³. \endcases We consider the case where the potential \(V\) is indefinite so that the Schrödinger operator \(-Δ+ V\) has a finite-dimensional negative space. Under suitable assumptions on the potential \(V\) and nonlinearity f(x,u), we prove the existence of nontrivial solutions via a local linking argument and Morse theory. Moreover, these solutions are shown to decay exponentially at infinity. Additionally, a ground state solution is obtained by minimization techniques. Finally, if \(f(x,u)\) is odd with respect to \(u\), we obtain an unbounded sequence of solutions using the symmetric mountain pass theorem.

Citations

• Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics

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