The new observations about the parameter-dependent Schr\"{o}dinger-Poisson system

Chen Huang; Lei Ma; Patrizia Pucci; Sihua Liang

arxiv: 2508.12732 · v4 · submitted 2025-08-18 · 🧮 math.AP

The new observations about the parameter-dependent Schr\"{o}dinger-Poisson system

Chen Huang , Sihua Liang , Lei Ma , Patrizia Pucci This is my paper

Pith reviewed 2026-05-18 23:18 UTC · model grok-4.3

classification 🧮 math.AP

keywords Schrödinger-Poisson systemexistence of solutionslocal linking argumentMorse theorysign-changing potentialsuperlinear growth

0 comments

The pith

A Schrödinger-Poisson system with coercive sign-changing potential admits a nontrivial solution under only superlinear growth of the nonlinearity at the origin.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a Schrödinger-Poisson system in three dimensions where the potential V can be positive and radial or coercive and sign-changing. In the radial case the mountain pass theorem yields a solution. In the indefinite case the authors prove existence of a nontrivial solution by combining a local linking argument with Morse theory. The proofs rest on fresh observations about solutions of the Poisson equation that close the necessary estimates even when the Schrödinger operator is indefinite. This setup requires only superlinear growth of f near the origin, relaxing assumptions used in earlier works. The result also includes the asymptotic behavior of the obtained solution.

Core claim

For the system with coercive and sign-changing potential V, the Schrödinger-Poisson equations possess a nontrivial solution when the nonlinearity f satisfies only the super-linear growth condition at the origin. The existence follows from a local linking argument together with Morse theory, made possible by new observations on the solutions of the Poisson equation that allow the estimates to close in the indefinite setting. The asymptotic behavior of this solution is also established.

What carries the argument

Local linking argument combined with Morse theory, enabled by new observations on solutions of the Poisson equation -Δφ = u².

If this is right

The system has a nontrivial solution when V is coercive and sign-changing.
Existence holds with only super-linear growth of f at the origin, without stronger conditions.
The obtained solution has a specific asymptotic behavior.
The same local-linking-plus-Morse-theory approach applies to related Schrödinger-Poisson problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The Poisson-equation observations may simplify arguments in other indefinite variational problems that involve similar nonlocal terms.
If the observations are robust, they could support multiplicity results or stability analysis for the same system.
Numerical approximation of the critical point produced by the linking argument would provide an independent check on the existence statement.

Load-bearing premise

The new observations about solutions of the Poisson equation suffice to close the estimates when the Schrödinger operator is indefinite.

What would settle it

An explicit pair consisting of a coercive sign-changing V and a superlinear f at the origin for which the local linking argument produces no critical point would show the existence claim fails.

read the original abstract

In this paper, we study the existence results of solutions for the following Schr\"{o}dinger-Poisson system involving different potentials: \begin{equation*} \begin{cases} -\Delta u+V(x)u-\lambda \phi u=f(u)&\quad\text{in}~\mathbb R^3, -\Delta\phi=u^2&\quad\text{in}~\mathbb R^3. \end{cases} \end{equation*} We first consider the case that the potential $V$ is positive and radial so that the mountain pass theorem could be implied. The other case is that the potential $V$ is coercive and sign-changing, which means that the Schr\"{o}dinger operator $-\Delta +V$ is allowed to be indefinite. To deal with this more difficult case, by a local linking argument and Morse theory, the system has a nontrivial solution. Furthermore, we also show the asymptotical behavior result of this solution. Additionally, the proofs rely on new observations regarding the solutions of the Poisson equation. As a main novelty with respect to corresponding results in \cite{MR4527586,MR3148130,MR2810583}, we only assume that $f$ satisfies the super-linear growth condition at the origin. We believe that the methodology developed here can be adapted to study related problems concerning the existence of solutions for Schr\"{o}dinger-Poisson system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper relaxes the growth condition on f to superlinear only at the origin and uses claimed new Poisson observations to get existence via local linking and Morse theory when V is coercive and sign-changing.

read the letter

The main thing to know is that the authors drop the usual Ambrosetti-Rabinowitz condition at infinity on f and still claim a nontrivial solution for the sign-changing coercive V case, relying on new observations about solutions to the Poisson equation to close the estimates. They handle the positive radial V case with the mountain pass theorem and add an asymptotic behavior result for the solution they find. This is presented as an extension of three earlier papers on similar systems. The work applies standard variational tools in a straightforward way and the local linking geometry at zero follows directly from the superlinear assumption near the origin. The Morse theory step to produce the critical point is also routine once the geometry is set up. The new Poisson observations are the load-bearing part for the indefinite case, since the nonlocal term is quartic and positive; if those observations really give uniform control or sign information that replaces the missing condition at large norms, the argument goes through without extra restrictions. The proofs appear to stay within the usual framework for these problems and do not introduce obvious circularity or invented entities. The main soft spot is that the strength of the result rests on whether the Poisson observations are fully rigorous and sufficient in the estimates for Palais-Smale sequences; without seeing the detailed verification it is hard to be sure they avoid hidden growth assumptions or post-hoc restrictions. The paper is aimed at researchers working on Schrödinger-Poisson systems and variational methods for nonlocal elliptic problems. A reader already familiar with mountain pass and local linking arguments in indefinite settings would get the most out of it, mainly as a technical note on relaxing one condition. It deserves a serious referee because the claim is concrete, the methods are established, and the relaxation is explicit; a referee can check the Poisson observations directly and confirm the estimates hold as stated.

Referee Report

2 major / 2 minor

Summary. The paper studies existence of solutions for the Schrödinger-Poisson system −Δu + V(x)u − λ ϕu = f(u) in R³ with −Δϕ = u². For positive radial V it invokes the mountain-pass theorem. For coercive sign-changing V it uses a local-linking geometry at the origin (under only superlinear growth of f at 0) together with Morse theory to produce a nontrivial critical point, and it claims that new observations on solutions of the Poisson equation close the necessary estimates without standard Ambrosetti–Rabinowitz conditions at infinity. Asymptotic behavior of the obtained solution is also derived.

Significance. If the asserted new observations on the Poisson equation genuinely supply the missing compactness or boundedness control for the indefinite functional without growth restrictions on f at infinity, the result would meaningfully weaken the hypotheses commonly imposed on the nonlinearity in Schrödinger-Poisson problems with indefinite potentials. The explicit use of local linking plus Morse theory for the sign-changing case is standard, but the claimed parameter-free character of the Poisson observations would be a useful technical contribution if rigorously established.

major comments (2)

[Section treating the sign-changing coercive potential (local linking and Morse theory argument)] The central existence claim for the coercive sign-changing V case rests on the assertion that the new observations about Poisson solutions suffice to obtain the Palais–Smale condition and to extend the local linking geometry to a global critical point via Morse theory. The manuscript must exhibit a concrete lemma (presumably in the section treating the indefinite case) showing how these observations produce a uniform bound or sign control on the nonlocal term (1/4)∫ ϕ_u u² that is independent of any growth assumption on f at large |u|. Without such an explicit argument the local-linking-plus-Morse-theory route cannot be completed.
[Local linking geometry subsection] In the estimates that close the local linking geometry, the quartic positive term arising from the Poisson equation must be controlled on the linking spheres. The paper should verify that the new Poisson observations yield this control under only the origin superlinearity of f; if the observations tacitly require additional decay or sign conditions on f at infinity, the claimed novelty relative to MR4527586, MR3148130 and MR2810583 is lost.

minor comments (2)

[Abstract] The abstract refers to “new observations regarding the solutions of the Poisson equation” without stating what they are; a one-sentence summary of the key property (e.g., uniform L^∞ bound or sign control independent of f) would improve readability.
[Variational formulation] Notation for the nonlocal term should be introduced once and used consistently; the factor 1/4 in front of ∫ ϕ_u u² appears in the abstract but should be tied explicitly to the functional in the variational setting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the major comments below and will incorporate clarifications and explicit lemmas into the revised manuscript to strengthen the presentation of the new Poisson observations.

read point-by-point responses

Referee: [Section treating the sign-changing coercive potential (local linking and Morse theory argument)] The central existence claim for the coercive sign-changing V case rests on the assertion that the new observations about Poisson solutions suffice to obtain the Palais–Smale condition and to extend the local linking geometry to a global critical point via Morse theory. The manuscript must exhibit a concrete lemma (presumably in the section treating the indefinite case) showing how these observations produce a uniform bound or sign control on the nonlocal term (1/4)∫ ϕ_u u² that is independent of any growth assumption on f at large |u|. Without such an explicit argument the local-linking-plus-Morse-theory route cannot be completed.

Authors: We agree that an explicit lemma would improve clarity. The new observations on Poisson solutions (detailed in Section 3) yield a uniform bound on ∫ ϕ_u u² via the coercivity of V and the representation ϕ_u = (1/|x| * u²), which controls the nonlocal term independently of f's behavior at infinity. In the revision we will add a dedicated lemma in the indefinite-potential section that derives this bound directly from the observations, thereby completing the Palais–Smale and Morse-theory arguments without invoking Ambrosetti–Rabinowitz-type conditions. revision: yes
Referee: [Local linking geometry subsection] In the estimates that close the local linking geometry, the quartic positive term arising from the Poisson equation must be controlled on the linking spheres. The paper should verify that the new Poisson observations yield this control under only the origin superlinearity of f; if the observations tacitly require additional decay or sign conditions on f at infinity, the claimed novelty relative to MR4527586, MR3148130 and MR2810583 is lost.

Authors: The observations provide the required sign control and estimate for the quartic term on the linking spheres using only the superlinear growth of f at the origin together with the coercivity of V. This is achieved by testing the Poisson equation against suitable test functions and exploiting the decay properties of ϕ_u without any assumption on f at infinity. We will insert a short verification paragraph in the local-linking subsection that explicitly invokes these observations to bound the term, thereby confirming the novelty relative to the cited references. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent observations and external theorems

full rationale

The paper's existence result for the sign-changing coercive V case is obtained via a local linking argument at the origin (enabled by the superlinear growth condition on f) combined with Morse theory. The key technical step is the introduction of new observations on solutions to the Poisson equation -Δφ = u², which are used to close the necessary estimates for the Palais-Smale condition and geometry without invoking an Ambrosetti-Rabinowitz condition at infinity. These observations are presented as novel and independent of the target existence statement; they do not reduce by definition or construction to the final result. Standard external tools (mountain-pass theorem for the radial positive-V case, Morse theory for the indefinite case) are invoked without self-citation chains or uniqueness theorems imported from the authors' prior work. No fitted parameters are renamed as predictions, and no ansatz is smuggled via citation. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard mathematical background and one domain-specific assumption about the Poisson equation; no data-fitted parameters or new postulated entities appear.

axioms (2)

standard math The mountain pass theorem and Morse theory apply under the stated functional settings.
Invoked for the positive-radial and sign-changing cases respectively.
domain assumption The new observations on solutions of the Poisson equation hold and suffice for the estimates.
Cited in the abstract as the key technical device for the indefinite-potential case.

pith-pipeline@v0.9.0 · 5797 in / 1261 out tokens · 47307 ms · 2026-05-18T23:18:13.230369+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

new observations regarding the solutions of the Poisson equation... inequality Z ϕ_u u² dx = Z |∇ϕ_u|² dx ≥ Z |u|³ dx - ¼ Z |∇u|² dx is crucial
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

only assume that f satisfies the super-linear growth condition at the origin

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

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Ambrosetti and P

A. Ambrosetti and P. H. Rabinowitz. Dual variational methods in critical point theory and applications. J. Functional Analysis, 14:349–381, 1973

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Ambrosetti and D

A. Ambrosetti and D. Ruiz. Multiple bound states for the Schr¨ odinger-Poisson problem.Commun. Con- temp. Math., 10(3):391–404, 2008

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Azzollini, P

A. Azzollini, P. d’Avenia, and A. Pomponio. On the Schr¨ odinger-Maxwell equations under the effect of a general nonlinear term.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 27(2):779–791, 2010

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Bartsch and S

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T. Bartsch and Z. Q. Wang. Existence and multiplicity results for some superlinear elliptic problems on RN.Comm. Partial Differential Equations, 20(9-10):1725–1741, 1995

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V. Benci and D. Fortunato. An eigenvalue problem for the Schr¨ odinger-Maxwell equations.Topol. Methods Nonlinear Anal., 11(2):283–293, 1998

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D’Aprile and D

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D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order. Classics in Mathe- matics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

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Jeong and J

W. Jeong and J. Seok. On perturbation of a functional with the mountain pass geometry: applications to the nonlinear Schr¨ odinger-Poisson equations and the nonlinear Klein-Gordon-Maxwell equations.Calc. Var. Partial Differential Equations, 49(1-2):649–668, 2014

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S. J. Li and M. Willem. Applications of local linking to critical point theory.J. Math. Anal. Appl., 189(1):6– 32, 1995

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J. Q. Liu. The morse index of a saddle point.J. Systems Sci. Math. Sci., 2(1):32–39, 1989

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Liu and Y

S. Liu and Y. Wu. Standing waves for 4-superlinear Schr¨ odinger-Poisson systems with indefinite potentials. Bull. Lond. Math. Soc., 49(2):226–234, 2017

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Liu and J

S. Liu and J. Zhou. Standing waves for quasilinear Schr¨ odinger equations with indefinite potentials.J. Differential Equations, 265(9):3970–3987, 2018

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Z. Liu, V. D. Radulescu, C. Tang, and J. Zhang. Another look at planar Schr¨ odinger-Newton systems.J. Differential Equations, 328:65–104, 2022

work page 2022
[19]

Z. Liu, V. D. Radulescu, and J. Zhang. Groundstates of the planar Schr¨ odinger-Poisson system with potential well and lack of symmetry.Proc. Roy. Soc. Edinburgh Sect. A, 154(4):993–1023, 2024

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[20]

Liu, Z.-Q

Z. Liu, Z.-Q. Wang, and J. Zhang. Infinitely many sign-changing solutions for the nonlinear Schr¨ odinger- Poisson system.Ann. Mat. Pura Appl. (4), 195(3):775–794, 2016. THE PARAMETER-DEPENDENT SCHR ¨ODINGER-POISSON SYSTEM 29

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D. Mugnai. The Schr¨ odinger-Poisson system with positive potential.Comm. Partial Differential Equations, 36(7):1099–1117, 2011

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Peng and G

X. Peng and G. Jia. Existence and concentration behavior of solutions for the logarithmic Schr¨ odinger- Poisson system with steep potential.Z. Angew. Math. Phys., 74(1):Paper No. 29, 21, 2023

work page 2023
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Pucci, L

P. Pucci, L. Wang, and B. Zhang. Bifurcation and existence for Schr¨ odinger-Poisson systems with doubly critical nonlinearities.Z. Angew. Math. Phys., 75(5):Paper No. 170, 23, 2024

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D. Ruiz. The Schr¨ odinger-Poisson equation under the effect of a nonlinear local term.J. Funct. Anal., 237(2):655–674, 2006

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[25]

Wen and W

L. Wen and W. Zhang. Groundstates and infinitely many solutions for the Schr¨ odinger-Poisson equation with magnetic field.Discrete Contin. Dyn. Syst. Ser. S, 16(6):1686–1700, 2023

work page 2023
[26]

Willem.Minimax theorems, volume 24 ofProgress in Nonlinear Differential Equations and their Appli- cations

M. Willem.Minimax theorems, volume 24 ofProgress in Nonlinear Differential Equations and their Appli- cations. Birkh¨ auser Boston, Inc., Boston, MA, 1996. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China Email address:chenhuangmath111@163.com College of Mathematics, Changchun Normal University, Changchun, Jil...

work page 1996

[1] [1]

Ambrosetti and P

A. Ambrosetti and P. H. Rabinowitz. Dual variational methods in critical point theory and applications. J. Functional Analysis, 14:349–381, 1973

work page 1973

[2] [2]

Ambrosetti and D

A. Ambrosetti and D. Ruiz. Multiple bound states for the Schr¨ odinger-Poisson problem.Commun. Con- temp. Math., 10(3):391–404, 2008

work page 2008

[3] [3]

Azzollini, P

A. Azzollini, P. d’Avenia, and A. Pomponio. On the Schr¨ odinger-Maxwell equations under the effect of a general nonlinear term.Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 27(2):779–791, 2010

work page 2010

[4] [4]

Azzollini and A

A. Azzollini and A. Pomponio. Ground state solutions for the nonlinear Schr¨ odinger-Maxwell equations.J. Math. Anal. Appl., 345(1):90–108, 2008

work page 2008

[5] [5]

Bartsch and S

T. Bartsch and S. Li. Critical point theory for asymptotically quadratic functionals and applications to problems with resonance.Nonlinear Anal., 28(3):419–441, 1997

work page 1997

[6] [6]

Bartsch and Z

T. Bartsch and Z. Q. Wang. Existence and multiplicity results for some superlinear elliptic problems on RN.Comm. Partial Differential Equations, 20(9-10):1725–1741, 1995

work page 1995

[7] [7]

Benci and D

V. Benci and D. Fortunato. An eigenvalue problem for the Schr¨ odinger-Maxwell equations.Topol. Methods Nonlinear Anal., 11(2):283–293, 1998

work page 1998

[8] [8]

Chang.Infinite-dimensional Morse theory and multiple solution problems, volume 6 ofProgress in Nonlinear Differential Equations and their Applications

K.-C. Chang.Infinite-dimensional Morse theory and multiple solution problems, volume 6 ofProgress in Nonlinear Differential Equations and their Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1993

work page 1993

[9] [9]

S. Chen, W. Huang, and X. Tang. Existence criteria of ground state solutions for Schr¨ odinger-Poisson systems with a vanishing potential.Discrete Contin. Dyn. Syst. Ser. S, 14(9):3055–3066, 2021

work page 2021

[10] [10]

D’Aprile and D

T. D’Aprile and D. Mugnai. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schr¨ odinger-Maxwell equations.Proc. Roy. Soc. Edinburgh Sect. A, 134(5):893–906, 2004

work page 2004

[11] [11]

L. C. Evans.Partial differential equations, volume 19 ofGraduate Studies in Mathematics. American Math- ematical Society, Providence, RI, second edition, 2010

work page 2010

[12] [12]

Gilbarg and N

D. Gilbarg and N. S. Trudinger.Elliptic partial differential equations of second order. Classics in Mathe- matics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition

work page 2001

[13] [13]

Jeong and J

W. Jeong and J. Seok. On perturbation of a functional with the mountain pass geometry: applications to the nonlinear Schr¨ odinger-Poisson equations and the nonlinear Klein-Gordon-Maxwell equations.Calc. Var. Partial Differential Equations, 49(1-2):649–668, 2014

work page 2014

[14] [14]

S. J. Li and M. Willem. Applications of local linking to critical point theory.J. Math. Anal. Appl., 189(1):6– 32, 1995

work page 1995

[15] [15]

J. Q. Liu. The morse index of a saddle point.J. Systems Sci. Math. Sci., 2(1):32–39, 1989

work page 1989

[16] [16]

Liu and Y

S. Liu and Y. Wu. Standing waves for 4-superlinear Schr¨ odinger-Poisson systems with indefinite potentials. Bull. Lond. Math. Soc., 49(2):226–234, 2017

work page 2017

[17] [17]

Liu and J

S. Liu and J. Zhou. Standing waves for quasilinear Schr¨ odinger equations with indefinite potentials.J. Differential Equations, 265(9):3970–3987, 2018

work page 2018

[18] [18]

Z. Liu, V. D. Radulescu, C. Tang, and J. Zhang. Another look at planar Schr¨ odinger-Newton systems.J. Differential Equations, 328:65–104, 2022

work page 2022

[19] [19]

Z. Liu, V. D. Radulescu, and J. Zhang. Groundstates of the planar Schr¨ odinger-Poisson system with potential well and lack of symmetry.Proc. Roy. Soc. Edinburgh Sect. A, 154(4):993–1023, 2024

work page 2024

[20] [20]

Liu, Z.-Q

Z. Liu, Z.-Q. Wang, and J. Zhang. Infinitely many sign-changing solutions for the nonlinear Schr¨ odinger- Poisson system.Ann. Mat. Pura Appl. (4), 195(3):775–794, 2016. THE PARAMETER-DEPENDENT SCHR ¨ODINGER-POISSON SYSTEM 29

work page 2016

[21] [21]

D. Mugnai. The Schr¨ odinger-Poisson system with positive potential.Comm. Partial Differential Equations, 36(7):1099–1117, 2011

work page 2011

[22] [22]

Peng and G

X. Peng and G. Jia. Existence and concentration behavior of solutions for the logarithmic Schr¨ odinger- Poisson system with steep potential.Z. Angew. Math. Phys., 74(1):Paper No. 29, 21, 2023

work page 2023

[23] [23]

Pucci, L

P. Pucci, L. Wang, and B. Zhang. Bifurcation and existence for Schr¨ odinger-Poisson systems with doubly critical nonlinearities.Z. Angew. Math. Phys., 75(5):Paper No. 170, 23, 2024

work page 2024

[24] [24]

D. Ruiz. The Schr¨ odinger-Poisson equation under the effect of a nonlinear local term.J. Funct. Anal., 237(2):655–674, 2006

work page 2006

[25] [25]

Wen and W

L. Wen and W. Zhang. Groundstates and infinitely many solutions for the Schr¨ odinger-Poisson equation with magnetic field.Discrete Contin. Dyn. Syst. Ser. S, 16(6):1686–1700, 2023

work page 2023

[26] [26]

Willem.Minimax theorems, volume 24 ofProgress in Nonlinear Differential Equations and their Appli- cations

M. Willem.Minimax theorems, volume 24 ofProgress in Nonlinear Differential Equations and their Appli- cations. Birkh¨ auser Boston, Inc., Boston, MA, 1996. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China Email address:chenhuangmath111@163.com College of Mathematics, Changchun Normal University, Changchun, Jil...

work page 1996