In this article, we investigate the existence of ground state solutions for the following generalized Schrödinger–Poisson system:
where is a parameter, , and . By combining the monotonicity trick with refined variational and analytical techniques, we establish the existence of ground state solutions for the above system under mild assumptions on and .
AguehM. (2006). Sharp Gagliardo–Nirenberg inequalities and mass transport theory. Journal of Dynamics and Differential Equations, 18, 1069–1093.
2.
AmbrosettiA.RuizD. (2008). Multiple bound states for the Schrödinger–Poisson problem. Communications in Contemporary Mathematics , 10, 391–404.
3.
AzzolliniA.d’AveniaP.LuisiV. (2013). Generalized Schrödinger–Poisson type systems. Communications on Pure and Applied Mathematics, 12, 867–879.
4.
AzzolliniA.d’AveniaP.PomponioA. (2010). On the Schrödinger–Maxwell equations under the effect of a general nonlinear term. Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 27(2), 779–791.
5.
AzzolliniA.PomponioA. (2008). Ground state solutions for the nonlinear Schrödinger–Maxwell equations. Journal of Mathematical Analysis and Applications, 345, 90–108.
6.
BenciV.FortunatoD. (1998). An eigenvalue problem for the Schrödinger–Maxwell equations. Topological Methods in Nonlinear Analysis, 11, 283–293.
7.
CeramiG.VairaG. (2010). Positive solutions for some non-autonomous Schrödinger–Poisson systems. Journal of Differential Equations, 248, 521–543.
8.
ChenS. T.TangX. H. (2016). Ground state sign-changing solutions for a class of Schrödinger–Poisson type problems in . Zeitschrift für Angewandte Mathematik und Physik, 67, 1–18.
9.
ChenC. Y.WuT. F. (2022). Positive solutions for nonlinear Schrödinger–Poisson systems with general nonlinearity. Nonlinear Differential Equations and Applications (NoDEA), 29, 58.
10.
CocliteG. M. (2003). A multiplicity result for the nonlinear Schrödinger–Maxwell equations. Communications on Applied Nonlinear Analysis, 7, 417–423.
11.
HeX. M. (2011). Multiplicity and concentration of positive solutions for the Schrödinger–Poisson equations. Zeitschrift für Angewandte Mathematik und Physik, 5, 869–889.
12.
JeanjeanL. (1999). On the existence of bounded Palais–Smale sequence and application to a Landesman–Lazer type problem set on . Proceedings of the Royal Society of Edinburgh. Section A, 129, 787–809.
13.
LiF.LiY.ShiJ. (2014). Existence of positive solutions to Schrödinger–Poisson type systems with critical exponent. Communications in Contemporary Mathematics, 16, 1450036.
14.
LiY.LiF.ShiJ. (2017). Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term. Calculus of Variations and Partial Differential Equations, 56, 134.
15.
LiF.ZhangQ. (2013). Existence of positive solutions to the Schrödinger–Poisson system without compactness conditions. Journal of Mathematical Analysis and Applications, 401, 754–762.
16.
LiebE. H. (1983). Sharp constants in the Hardy–Littlewood–Sobolev inequality and related inequalities. Annals of Mathematics, 118, 349–374.
17.
RuizD. (2006). The Schrödinger–Poisson equation under the effect of a nonlinear local term. Journal of Functional Analysis, 237, 655–674.
18.
SeokJ. (2013). On nonlinear Schrödinger–Poisson equations with general potentials. Journal of Mathematical Analysis and Applications, 401, 672–681.
19.
SunJ. J.MaS. W. (2016). Ground state solutions for some Schrödinger–Poisson systems with periodic potentials. Journal of Differential Equations, 260, 2119–2149.
20.
TangX. H.ChenS. T. (2017). Ground state solutions of Nehari–Pohoaev type for Schrödinger–Poisson problems with general potentials. Discrete and Continuous Dynamical Systems, 37, 4973–5002.
21.
WillemM. (1996). Minimax theorems, progress in nonlinear differential equations and their applications (Vol. 24). Birkhäuser Boston Inc.
22.
ZhaoL. G.ZhaoF. K. (2008). On the existence of solutions for the Schrödinger–Poisson equations. Journal of Mathematical Analysis and Applications, 346, 155–169.