Ground state solutions for the nonlinear Schrödinger‐Poisson systems with sum of periodic and vanishing potentials

Abstract

We study the existence of ground state solutions for the following Schrödinger-Poisson equations:whereis the sum of a periodic potential V p and a localized potential V loc and f satisfies the subcritical or critical growth. Although the Nehari-type monotonicity assumption on f is not satisfied in the subcritical case, we obtain the existence of a ground state solution as a minimizer of the energy functional on Nehari manifold. Moreover, we show that the existence and nonexistence of ground state solutions are dependent on the sign of V loc .

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

“…The precise conditions on will be given later on. Problem possesses many physical motivations; eg, it arises from approximation of the Hartree‐Fock equation that describes a quantum mechanical of many particles; see, for example, previous studies …”

“…Problem (1) possesses many physical motivations; eg, it arises from approximation of the Hartree-Fock equation that describes a quantum mechanical of many particles; see, for example, previous studies. [1][2][3][4][5][6][7][8][9] Over the past few decades, much attention has been drawn to the study of stationary solutions (x, t) = e −i t u(x) to (1), where ∈ R and u ∶ R 3 → R. Then (1) is reduced to be the following system…”

Existence and limiting behavior of minimizers for attractive Schrödinger‐Poisson systems with periodic potentials

“…and the strict inequality holds on a set of positive measure; Sun and Ma 21 for periodic external potentials; Xie et al 22 for a sum of periodic and vanishing potentials; and Zeng and Zhang 23 for trapping external potential, ie,…”

“…The existence of standing waves as well as their stability or instability in the case of repulsive interaction has been studied in many works; see, for instance, previous studies for nonexternal potential, ie, ; Ambrosetti and Ruiz for radially symmetric external potential; Azzollini for external potentials satisfying and the strict inequality holds on a set of positive measure; Sun and Ma for periodic external potentials; Xie et al for a sum of periodic and vanishing potentials; and Zeng and Zhang for trapping external potential, ie, In the case of attractive interaction , the existence of standing waves and its limiting behavior has been considered in recent works. …”

Existence and multiplicity of nontrivial solutions for Schrödinger-Poisson systems on bounded domains

“…Recently, Schrödinger-Poisson systems on unbounded domains or on the whole space R N have attracted a lot of attention. Many solvability conditions on the nonlinearity have been given to obtain the existence and multiplicity of solutions for Schrödinger-Poisson systems in R N , we refer the readers to [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and references therein.…”