Positive multi-peak solutions for a logarithmic Schrodinger equation

Peng Luo; Yahui Niu

REVIEW 3 major objections 4 minor 29 references

Positive multi-peak solutions for a logarithmic Schrodinger equation

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that for any finite set of non-degenerate critical points of the potential, the logarithmic Schrödinger equation admits positive k-peak solutions for sufficiently small ε, and any two such solutions concentrating at the…

desk verdict The first reduction-based attack on logarithmic Schrödinger multi-peak existence has a false convolution estimate at its core, so the main theorems are unproven; still worth a referee's time. read the letter →

arxiv 1908.02970 v1 pith:TQZQQ73K submitted 2019-08-08 math.AP

Peng Luo , Yahui Niu This is my paper · ORCID

classification math.AP MSC 35B2535J1035J60

keywords logarithmicSchrödingerequationmulti-peaksolutionsLyapunov–SchmidtreductionweightednormlocalPohozaevidentitiesuniquenesssingularperturbation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the singularly perturbed logarithmic Schrödinger equation $-\varepsilon^2\Delta u + V(x)u = u \log u^2$ has positive multi-peak solutions concentrated at any prescribed non-degenerate critical points of $V$, for $\varepsilon$ small. This is the first Lyapunov–Schmidt reduction for a logarithmic Schrödinger equation; the difficulty is that the logarithmic nonlinearity makes the variational functional fail to be $C^1$, and the usual remainder estimate fails in $H^1$. The authors introduce a weighted sup norm, the star norm, that measures a correction $\phi$ against the sum of the Gaussian-shaped bumps used as building blocks, and run the contraction argument in that norm. They then use local Pohozaev identities to select the peak locations and to obtain local uniqueness. The result is a general reduction scheme for logarithmic nonlinearities and a uniqueness theorem for multi-peak bound states.

What carries the argument

The load-bearing object is the weighted norm $\|\phi\|_*=\sup_x (\sum_{j=1}^k e^{-|x-y_j|^2/(2\varepsilon^2)})^{-1}|\phi(x)|$. It replaces the ordinary $H^1$ norm as the metric for the contraction step, making the nonlinear remainder $R_\varepsilon(\phi)$ contractive, $\|R_\varepsilon(\phi)\|_*=O(\|\phi\|_*^2)$, which is false for the $H^1$ norm. Around this sit four standard pieces: the explicit ground state $U(x)=e^{(\omega+N-|x|^2)/2}$ and the non-degeneracy of its linearization, whose kernel is spanned by $\partial U/\partial x_j$; the projection $P_\varepsilon$ onto the orthogonal complement of that kernel; the invertibility of $P_\varepsilon L_\varepsilon$ with a uniform bound; and a family of local Pohozaev identities that convert the vanishing of the finite-dimensional Lagrange multipliers into the equations $\nabla V(y_j)=O(|\log\varepsilon|^{-(1-\theta)})$, solvable because $V$ has non-degenerate critical points. The same identities, combined with a blow-up analysis and the maximum principle, drive the uniqueness proof.

What would settle it

Compute the integral $I(x)=\int_{\mathbb{R}^N}|z-x|^{2-N}e^{-|z-y|^2/(2\varepsilon^2)}\,dz$ for $|x-y|\gg\varepsilon$. For $N\ge 3$ it behaves like $(2\pi\varepsilon^2)^{N/2}|x-y|^{2-N}$, an algebraic decay, not the exponential barrier $\varepsilon^4|x-y|^{-2}e^{-|x-y|^2/(2\varepsilon^2)}$ used in estimate (3.5). The contradiction would show that the bounds on $u_1$ and $u_2$ in Proposition 3.2 do not follow from the stated argument.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for $N\ge 3$ and a potential $V$ satisfying (V1) and (V2), equation (1.1) has, for sufficiently small $\varepsilon$, a positive $k$-peak solution whose peaks tend to the non-degenerate critical points $\xi_1,\dots,\xi_k$, and Theorem 1.2: any two such positive solutions concentrating at the same points coincide for small $\varepsilon$. The construction writes the solution as $\sum_{j=1}^k U_{\varepsilon,y_j}+\phi$, where each $U_{\varepsilon,y_j}$ is the explicit Gaussian-type solution of the rescaled limiting equation, and solves for $\phi$ in a subspace orthogonal to the approximate kernel. The paper also proves sharp concentration information: $|y_{\varepsilon,j}-\xi_j|=o(\varepsilon)$ and $\|\phi_\varepsilon\|_\varepsilon=O(\varepsilon^{N/2+2})$.

Load-bearing premise

The proof rests on the assumption that the influence of each Gaussian-shaped bump on far-away points dies off exponentially fast; if that convolution estimate fails, the weighted contraction argument that produces the correction term does not close.

Editorial extensions

If this is right

For any finite set of non-degenerate critical points of $V$, equation (1.1) admits a positive solution with a peak near each point for all sufficiently small $\varepsilon$.
In the single-peak case with a strict global minimum point, the ground state of the equation is unique.
The Lyapunov–Schmidt reduction is now a viable tool for logarithmic nonlinearities, not just polynomial or saturable ones, once the correction is measured in the weighted norm.
The $o(\varepsilon)$ location of the peaks shows the constructed solutions are not merely abstract limits but concentrate with sharp precision.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same weighted-norm reduction should extend to logarithmic equations with fractional Laplacians or logarithmic Schrödinger–Newton systems, provided the convolution kernel estimate is re-derived for the new Green's function.
One could test the predicted $o(\varepsilon)$ peak-location rate numerically in one dimension by continuing the two solutions in $\varepsilon$; agreement would support the local-uniqueness conclusion.
The Pohozaev treatment of the finite-dimensional problem suggests the existence result may persist for potentials whose critical points are saddles of any index, a regime where minimization arguments fail.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: Theorems 1.1 and 1.2 are not proved. The load-bearing step is the convolution estimate (3.5) in Proposition 3.2, and it is wrong. The Newtonian potential of a Gaussian does not decay exponentially; for |x−y_j| ≫ ε it behaves like C_N ε^N / |x−y_j|^{N−2}, while the barrier w(x)= ε^4 |x−y_j|^{-2} e^{−|x−y_j|^2/(2ε^2)} is exponentially small. So the ratio LHS/w is unbounded. In fact the pointwise inequality −Δw ≥ C e^{−|x−y_j|^2/(2ε^2)} is itself false at large distances because −Δw turns negative. Either way, the estimates on u1 and u5 in Proposition 3.2 do not follow, the contraction in the weighted *-norm doesn't close, and the existence theorem collapses. Since Theorem 1.2 assumes those solutions, it goes down with it.

What is genuinely new: this is indeed the first Lyapunov–Schmidt reduction attempt for the logarithmic Schrödinger equation. The weighted norm ||·||_* is a reasonable idea—the log nonlinearity does create a singularity when the base concentration sum is small, and some weighted sup norm is natural. The use of local Pohozaev identities in Section 5 to avoid a global variational argument is standard in the modern literature but sensibly applied. The invertibility proof for P_ε L_ε and the estimates on l_ε and R_ε in Section 2 look standard and plausible.

Where the soft spots are, in proportion: the one fatal flaw is the estimate above. It is not a typo; the claimed inequality is used in exactly the place the new norm is supposed to work. Away from that, I have concerns about the hand-waving in Lemma 5.2 and the maximum-principle argument at the end of Theorem 1.2, but those are secondary. The citation pattern is fair; the paper cites the relevant variational, penalization, and local-Pohozaev literature.

Who this is for: an expert in singularly perturbed elliptic PDEs or logarithmic Schrödinger equations. It shows the shape of a potential reduction proof, but as submitted it is not a reliable source.

Recommendation: I would not desk-reject—the claim of a first reduction is important and the error is specific enough to be checkable. Send it to a referee with strong harmonic analysis background. My own recommendation to the editor would be major revision at best; the convolution estimate is central and needs a genuinely different argument, not a tweak.

Referee Report

3 major / 4 minor

Summary. The manuscript studies the singularly perturbed logarithmic Schrödinger equation (1.1) in R^N, N≥3, with a bounded positive potential V satisfying (V1) and having k non-degenerate critical points (V2). The authors propose to construct multi-peak positive solutions by Lyapunov-Schmidt reduction, using the unique positive solution U of the limiting equation as building blocks and introducing a weighted sup-norm ||·||_* (defined in (1.8)) to control the remainder R_ε(φ), whose standard H^1 estimate fails because of the logarithmic nonlinearity. The main existence theorem (Theorem 1.1) asserts the existence of k-peak solutions concentrating at the critical points for small ε. A second theorem (Theorem 1.2) asserts local uniqueness of such positive solutions via local Pohozaev identities. The paper contains the standard two-step reduction: invertibility of the linearized operator (Proposition 2.1), estimates of the error terms (Lemmas 2.2–2.3 and 3.1), a contraction-mapping fixed point (Proposition 3.3), and a finite-dimensional reduction using Pohozaev identities to locate the peaks (Theorem 4.1). Section 5 then proves local uniqueness by comparing two solutions through their normalized difference.

Significance. If the results were correct, they would provide the first Lyapunov-Schmidt reduction for a logarithmic Schrödinger equation, together with a local uniqueness theorem for multi-peak solutions, a significant advance over purely variational existence results. The authors correctly identify the main technical difficulty (the non-Lipschitz nature of the logarithmic term) and propose an interesting weighted norm to address it. The paper is ambitious, the structure is clear, and the use of local Pohozaev identities for the finite-dimensional reduction is well motivated. However, the central analytical estimate on which the new norm is designed to work is false, so the main theorem is not established.

major comments (3)

[Section 3, Eq. (3.5)] The convolution estimate (3.5) is false. For w(x)=ε^4 |x-y_j|^{-2} e^{-|x-y_j|^2/(2ε^2)}, the claim that -Δw ≥ C e^{-|x-y_j|^2/(2ε^2)} and hence ∫ e^{-|z-y_j|^2/(2ε^2)} |z-x|^{2-N} dz ≤ C w(x) is incorrect. For |x-y_j| ≫ ε, the Newtonian potential of a Gaussian has the algebraic asymptotic C_N ε^N |x-y_j|^{2-N}, whereas w decays exponentially in |x-y_j|^2/(2ε^2). At |x-y_j| = Rε the ratio of the true convolution to w is of order R^{2-N} e^{R^2/2}, which is unbounded as R→∞. Consequently, the bounds on u1 in (3.6)–(3.11), on u2 in (3.12)–(3.16), on u3 in (3.17)–(3.19), and on u5 in (3.21)–(3.27) do not follow. Proposition 3.2's conclusion ||u||_* ≤ |ln ε|^{-(1-θ)} is therefore unsupported, and the contraction-mapping argument in Proposition 3.3, which relies on Proposition 3.2, collapses. This is a load-bearing error in the proof of Theorem 1.1.
[Section 2, Eq. (2.7) and Proposition 2.1] Formula (2.7) does not define a projection onto E_{ε,y}. For a general u∈H^1, u - Σ_{j,i} ⟨u, ∂U_{ε,y_j}/∂x_i⟩ ∂U_{ε,y_j}/∂x_i is not orthogonal to span{∂U_{ε,y_j}/∂x_i} unless the set {∂U/∂x_i} is orthonormal, which it is not (the Gram matrix has off-diagonal entries of order ε^N). The proof of Proposition 2.1 uses the identity ⟨P_ε L φ, ψ⟩ = ⟨L φ, ψ⟩ for ψ∈E_{ε,y}; this identity does hold with the stated formula for ψ∈E because ⟨∂U_i, ψ⟩ = 0, but P_ε u does not lie in E_{ε,y}, so the reduction equation (3.1) is not an equation in E_{ε,y}. The argument can likely be repaired by using the orthogonal projection with the inverse Gram matrix, but as written this is a gap in the foundational invertibility result on which the whole reduction rests.
[Section 5, proof of Theorem 1.2] The proof of Theorem 1.2 assumes without proof that every positive solution u_ε^{(i)} concentrated at ξ_1,...,ξ_k has the form u_ε^{(i)} = Σ_{j=1}^k U_{ε,y_{ε,j}^{(i)}} + φ_ε^{(i)} with the estimates from Lemma 5.4 and Proposition 5.3. Proposition 5.3 is proved only for solutions produced by the Lyapunov-Schmidt construction in Theorem 4.1. For the local-uniqueness claim, one must first establish such an expansion for arbitrary positive solutions concentrated at the given points; no argument or citation is provided. Thus Theorem 1.2 is not supported even if the expansion for the constructed solutions were valid.

minor comments (4)

[Title and Abstract] The title and abstract contain several typographical errors (e.g., 'Schrodinger' for 'Schrödinger', 'Lyap unov' for 'Lyapunov', 'S chrodinger'); these should be corrected.
[Section 2, Lemma 2.3] The notation ⟨R(φ),η⟩_ε is inconsistent with the definition of the H_ε inner product; the estimate in the proof is for ∫ R(φ)η. This should be clarified, as the symbol ⟨·,·⟩_ε usually denotes the H_ε inner product introduced earlier.
[Section 3, Eq. (3.24)] The estimate in (3.24) is ambiguous: 'O( ε/R^{N-2} e^{R^2} )' should specify whether the exponential factor is in the numerator or denominator, and the power of R in the denominator should be checked against the computation.
[Section 5, Eq. (5.14)] The expansion (5.14) for U_{ε,y^{(1)}} - U_{ε,y^{(2)}} is schematic; a rigorous statement would use the mean value theorem with explicit bounds on the second derivatives of U. As written, the factor (y^{(1)}-y^{(2)})/ε is correctly identified as o(1) by (5.2), but the pointwise validity of the displayed formula in the support of the gradient should be justified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction solves an independent fixed-point problem; self-citations are technical, not definitional.

full rationale

The derivation chain is a standard Lyapunov-Schmidt reduction. The approximate solution U_{ε,y_j} is taken from the unique nondegenerate ground state of the limiting equation (cited to [14], an external source), and the remainder φ is obtained as the fixed point of φ = (P_ε L_ε)^{-1}(l_ε + R_ε(φ)) in the weighted space defined by (1.8)–(1.9). No parameter appearing in the conclusion is fitted to a quantity containing the conclusion: l_ε is bounded in Lemma 2.2 in terms of ∇V(y_j) and ε, and the reduced equations (4.2) are solved by the nondegeneracy assumption (V2), not by assuming the k-peak solution exists. Theorem 1.2 uses the Pohozaev identities proved in Proposition 5.1, a blow-up argument, and exponential decay estimates. Lemma 5.2 is imported by reference to [19] and the exterior maximum-principle argument to [8, Prop. 3.5]; these are prior papers (one author overlapping) but concern different problems or standard estimates and do not assume the present Theorem 1.1 or 1.2, so the citations are independent support rather than circularity. The possible defect in the convolution bound (3.5) would undermine Proposition 3.2 as a mathematical error, but that is a correctness question: (3.5) is not equivalent by construction to the existence or uniqueness conclusion, and no norm was defined in terms of the final peak locations in a way that makes the desired estimate an identity. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No parameters are fitted; the proof is analytic. The main axioms are standard non-degeneracy of the limiting profile and the unstated C^2 regularity of V. The critical technical input is the asserted convolution bound (3.5), which is flagged as invalid.

assumptions (4)

domain assumption The limiting profile U is the unique positive solution and is non-degenerate in H^1.
Cited to [14]; used to identify the kernel of the linearized operator in Proposition 2.1 and Proposition 5.5.
domain assumption The potential V is C^2 in a neighborhood of the critical points; the paper states C^1 but (V2) uses the Hessian determinant.
Section 1, assumptions (V1)-(V2). The regularity assumption is inconsistent as written.
ad hoc to paper The weighted norm method relies on the convolution bound (3.5), which is asserted but not valid.
Section 3, equation (3.5) in the proof of Proposition 3.2. This is the load-bearing estimate.
domain assumption Local Pohozaev identities from [22] apply to the logarithmic nonlinearity.
Section 5, Proposition 5.1. The identities are derived by multiplying the equation by ∂u/∂x_i and integrating by parts.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Positive multi-peak solutions for a logarithmic Schrodinger equation." pith.science (2026). https://pith.science/paper/TQZQQ73K

@misc{pith2026190802970,
  author       = {Pith},
  title        = {Pith review of: Positive multi-peak solutions for a logarithmic Schrodinger equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQZQQ73K}},
  note         = {Machine review of arXiv:1908.02970}
}

abstract

In this manuscript, we consider the logarithmic Schr\"{o}dinger equation \begin{eqnarray*} -\varepsilon^2\Delta u+V(x)u=u\log u^{2},\,\,\,u>0, & \text{in}\,\,\,\mathbb{R}^{N}, \end{eqnarray*} where $N\geq3$, $\varepsilon>0$ is a small parameter. Under some assumptions on $V(x)$, we show the existence of positive multi-peak solutions by Lyapunov-Schmidt reduction. It seems to be the first time to study singularly perturbed logarithmic Schr\"{o}dinger problem by reduction. And here using a new norm is the crucial technique to overcome the difficulty caused by the logarithmic nonlinearity. At the same time, we consider the local uniqueness of the multi-peak solutions by using a type of local Pohozaev identities.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

[1]

Al ves, D.C

C.O. Al ves, D.C. de Morais Filho, Existence and concentration of positive solutions for a Schrödinger logarithmic equation. Z. Angew. Math. Phys. 69 (2018), 144, 22 pp

work page 2018
[2]

Multiple positive solutions for a Schr\"{o}dinger logarithmic equation

C.O. Al ves, C. Ji, Multiple positive solutions for a Schrödinger logarithmic equation. arXiv:1901.10329v1

work page Pith review arXiv 1901
[3]

Ardila, Orbital stability of Gausson solutions to logarithmic Schr ödinger equations

A.H. Ardila, Orbital stability of Gausson solutions to logarithmic Schr ödinger equations. Electron. J. Diﬀerential Equations 335 (2016), 1-9

work page 2016
[4]

Ardila, Existence and stability of standing waves for nonlinear fra ctional Schrödinger equation with logarithmic nonlinearity

A.H. Ardila, Existence and stability of standing waves for nonlinear fra ctional Schrödinger equation with logarithmic nonlinearity. Nonlinear Analysis 155 (2107), 52-64

work page
[5]

Bahri, Critical Points at Inﬁnity in Some Variational Problems

A. Bahri, Critical Points at Inﬁnity in Some Variational Problems. Pitman Res. Notes Math. Ser., vol. 182, Longman Scientiﬁc and Technical, 1989

work page 1989
[6]

Bialynicki-Birula, J

I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics. Ann. Phys 100 (1976), 62-93

work page 1976
[7]

Buljan, A

H. Buljan, A. Siber, M. Soljacic, T. Schw ar tz, M. Segev, D.N. Christodoulides, Inco- herent white light solitons in logarithmically saturable n oninstantaneous nonlinear media. Phys. Rev. E 68 (2003), Article ID 036607

work page 2003
[8]

D. Cao, S. Li, P. Luo, Uniqueness of positive bound states with multi-bump for non linear Schrödinger equations. Calc Var Partial Diﬀerential Equat ions, 2015, 54:4037-4063

work page 2015

Show all 29 references

[9]

Cao, E.S

D. Cao, E.S. Noussair, S. Yan , Solutions with multiple peaks for nonlinear elliptic equat ions. Proc. Royal Soc. Edinburgh 129(1999), 235-264

1999
[10]

D. Cao, S. Peng, S. Yan , Singularly perturbed methods for nonlinear elliptic probl ems. Cambridge University Press, 2019

2019
[11]

Cazena ve, Stable solutions of the logarithmic Schrödinger equation

T. Cazena ve, Stable solutions of the logarithmic Schrödinger equation. Nonlinear. Anal. 7 (1983), 1127-1140. A LOGARITHMIC SCHRÖDINGER EQUATIONS 29

1983
[12]

Cazena ve, A

T. Cazena ve, A. Haraux, Equations d’évolution avec non-linéarité logarithmique. Ann. Fac. Sci. Toulouse Math. 2 (1980), 21-51

1980
[13]

Cazena ve, P.L

T. Cazena ve, P.L. Lions, Orbital stability of standing waves for some nonlinear Schr ödinger equa- tions. Comm. Math. Phys. 85 (1982), 549-561

1982
[14]

d’A venia, E

P. d’A venia, E. Montefusco, M. Squassina, On the logarithmic Schrödinger equation . Commun. Contemp. Math. 16 (2014), 1350032, 15pp

2014
[15]

d’A venia, M

P. d’A venia, M. Squassina, M. Zenari, Fractional logarithmic Schrödinger equations . Math. Methods Appl. Sci. 38 (2015), 5207-5216

2015
[16]

De Mar tino, M

S. De Mar tino, M. F alanga, C. Godano, G. Lauro, Logarithmic Schrödinger-like equation as a model for magma transport. Europhys Lett. 63 (2003), 472-475

2003
[17]

Guerrero, J.L

P. Guerrero, J.L. López, J. Nieto, Global H 1solvability of the 3D logarithmic Schrödinger equa- tion. Nonlinear Anal. Real World Appl., 11 (2010), 79-87

2010
[18]

Hefter, Application of the nonlinear Schrödinger equation with a lo garithmic inhomogeneous term to nuclear physics

E.F. Hefter, Application of the nonlinear Schrödinger equation with a lo garithmic inhomogeneous term to nuclear physics. Phys. Rev. 32 (1985), 1201-1204

1985
[19]

P. Luo, S. Peng, C. W ang Uniqueness of positive solutions with Concentration for th e Schrödinger- Newton problem. arXiv:1703.00777

arXiv
[20]

C. Ji, A. Szulkin, A logarithmic Schrödinger equation with asymptotic condit ions on the potential. J. Math. Anal. Appl., 437 (2016), 241-254

2016
[21]

E.H. Lieb, M. Loss, Gradute Studies in Mathematics. AMS, Providence, Rhodeisland. (2001)

2001
[22]

S. Peng, C. W ang, S. Yan Construction of solutions via local Pohozaev identities. J. Funct. Anal. 274 (2018), 2606šC2633

2018
[23]

Shuai, Multiple solutions for logarithmic Schrödinger equations

W. Shuai, Multiple solutions for logarithmic Schrödinger equations
[24]

Squassina, A

M. Squassina, A. Szulkin, Multiple solutions to logarithmic Schrödinger equations w ith periodic potential. Calc. Var. Partial Diﬀerential Equations 54 (2015), 585-597

2015
[25]

Squassina, A

M. Squassina, A. Szulkin, Erratum to: Multiple solutions to logarithmic Schrödinger equations with periodic potential. Calc. Var. Partial Diﬀerential Equations 56 (2017), 4 pp

2017
[26]

Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems

A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 77šC109

1986
[27]

Tanaka, C

K. Tanaka, C. Zhang, Multi-bump solutions for logarithmic Schr?dinger equatio ns. Calc. Var. Partial Diﬀerential Equations 56 (2017), 35 pp

2017
[28]

Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: O rigin of time and observational consequences

K.G. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: O rigin of time and observational consequences. Grav. Cosmol. 16(2010), 288-297

2010
[29]

Zloshchastiev, pontaneous symmetry breaking and mass generation as built- in phenomena in logarithmic nonlinear quantum theory

K.G. Zloshchastiev, pontaneous symmetry breaking and mass generation as built- in phenomena in logarithmic nonlinear quantum theory. Acta Physica Polonica B 42 (2011), 261-292. (Peng Luo) School of Ma thema tics and Sta tistics and Hubei Key Labora to r y of Ma the- ma tical S...

2011

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Al ves, D.C

C.O. Al ves, D.C. de Morais Filho, Existence and concentration of positive solutions for a Schrödinger logarithmic equation. Z. Angew. Math. Phys. 69 (2018), 144, 22 pp

work page 2018

[2] [2]

Multiple positive solutions for a Schr\"{o}dinger logarithmic equation

C.O. Al ves, C. Ji, Multiple positive solutions for a Schrödinger logarithmic equation. arXiv:1901.10329v1

work page Pith review arXiv 1901

[3] [3]

Ardila, Orbital stability of Gausson solutions to logarithmic Schr ödinger equations

A.H. Ardila, Orbital stability of Gausson solutions to logarithmic Schr ödinger equations. Electron. J. Diﬀerential Equations 335 (2016), 1-9

work page 2016

[4] [4]

Ardila, Existence and stability of standing waves for nonlinear fra ctional Schrödinger equation with logarithmic nonlinearity

A.H. Ardila, Existence and stability of standing waves for nonlinear fra ctional Schrödinger equation with logarithmic nonlinearity. Nonlinear Analysis 155 (2107), 52-64

work page

[5] [5]

Bahri, Critical Points at Inﬁnity in Some Variational Problems

A. Bahri, Critical Points at Inﬁnity in Some Variational Problems. Pitman Res. Notes Math. Ser., vol. 182, Longman Scientiﬁc and Technical, 1989

work page 1989

[6] [6]

Bialynicki-Birula, J

I. Bialynicki-Birula, J. Mycielski, Nonlinear wave mechanics. Ann. Phys 100 (1976), 62-93

work page 1976

[7] [7]

Buljan, A

H. Buljan, A. Siber, M. Soljacic, T. Schw ar tz, M. Segev, D.N. Christodoulides, Inco- herent white light solitons in logarithmically saturable n oninstantaneous nonlinear media. Phys. Rev. E 68 (2003), Article ID 036607

work page 2003

[8] [8]

D. Cao, S. Li, P. Luo, Uniqueness of positive bound states with multi-bump for non linear Schrödinger equations. Calc Var Partial Diﬀerential Equat ions, 2015, 54:4037-4063

work page 2015