Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials

Guozhen Lu; Lu Chen; Maochun Zhu

REVIEW 2 major objections 7 minor 53 references

Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials

T0 review · 2 major / 7 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 ground-state existence for the bi-harmonic equation with critical exponential growth in $\mathbb{R}^4$, under constant and Rabinowitz-type trapping potentials, and supplies the compactness criteria that carry the proof.

desk verdict The H²(R⁴) compactness criteria and the constant-potential ground state are solid, but the trapping-potential theorem rests on an unjustified strict comparison m_V < m_∞. read the letter →

arxiv 1909.01952 v1 pith:LYMZBPCE submitted 2019-09-04 math.AP math.CA

Lu Chen , Guozhen Lu , Maochun Zhu This is my paper · ORCID

classification math.APmath.CA MSC 46E3535J9126D10

keywords Rabinowitzpotentialgroundstatesolutionsbi-harmonicequationscriticalexponentialgrowthAdamsinequalitiesPohozaevidentityNeharimanifoldFourierrearrangement

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 establishes the existence of ground-state solutions for the bi-harmonic equation $(-\Delta)^2u+V(x)u=\lambda u e^{2|u|^2}$ in $\mathbb{R}^4$, where the nonlinearity has critical exponential growth. For constant potentials $V(x)=\gamma$, it proves a radial ground state exists for every $\lambda\in(0,\gamma)$; for Rabinowitz-type trapping potentials, satisfying $0<\lambda

What carries the argument

The argument is carried by four objects. The Adams inequality with exact growth in $\mathbb{R}^4$ (and its sharp Sobolev-norm version) fixes the critical constant $32\pi^2$ and controls exponential integrals. The Fourier rearrangement of the solution, replacing the unavailable pointwise symmetrization for second-order derivatives, turns an arbitrary minimizing sequence into a radial one. The Pohozaev identity defines the manifold on which the constrained infimum is searched for the constant-potential problem. Finally, the Nehari manifold $N_V$ and its limiting counterpart $N_\infty$ are compared; the strict inequality $m_V<m_\infty$ is the mechanism that prevents the loss of compactness and yields the ground state for the trapping potential.

What would settle it

Compute (analytically or numerically) the ratio $m_V/m_\infty$ for the potential $V(x)=\gamma+\varepsilon(1+|x|^2)^{-1}$ in $\mathbb{R}^4$, which obeys $\inf V<\gamma=\lim V$ but exceeds $\gamma$ near the origin; finding $m_V\ge m_\infty$ for any $\varepsilon>0$ would falsify Lemma 6.2 and thereby the proof of Theorem 2.7, while a verification that the strict inequality persists would support the theorem.

Watch

Extended reading notes

Core claim

The central claim is that the critical-exponential bi-harmonic problem in $\mathbb{R}^4$ admits ground states in two settings where the usual compactness mechanisms are absent. The paper first proves a boundedness criterion: if $g:\mathbb{R}\to[0,\infty)$ satisfies $\lim_{t\to\infty}|t|^2e^{-|t|^2/K}g(t)<\infty$ and $\lim_{t\to0}|t|^{-2}g(t)<\infty$, then $\int_{\mathbb{R}^4}g(u)\,dx\le C\int_{\mathbb{R}^4}|u|^2\,dx$ whenever $\|\Delta u\|_2^2\le32\pi^2K$, and an analogous compactness criterion with the limits equal to zero, valid for radial sequences. Using these, the paper shows that for $V(x)\equiv\gamma$ the transformed functional has a radial minimizer on the Pohozaev manifold, yielding a radial ground state for each $\lambda\in(0,\gamma)$. For a trapping potential, the decisive step is the strict inequality $m_V<m_\infty$ between the Nehari-manifold energies of the original and limiting equations; this inequality excludes vanishing and concentration, so the ground state is attained as a non-radial solution.

Load-bearing premise

The proof of the strict energy inequality $m_V<m_\infty$ assumes the potential satisfies $V(x)\le\gamma$ for all $x$, whereas the theorem's hypotheses only require $\inf V<\lim_{|x|\to\infty}V=\gamma$; if $V$ rises above $\gamma$ somewhere, that inequality (Lemma 6.2) is not justified.

Editorial extensions

If this is right

For every $\gamma>0$ and $\lambda\in(0,\gamma)$, the constant-potential equation $(-\Delta)^2u+\gamma u=\lambda u e^{2|u|^2}$ admits a radial ground state in $\mathbb{R}^4$ (Theorem 2.6).
For any continuous trapping potential with $0<\lambda<\inf V<\lim V<\infty$, a non-radial ground state exists in $\mathbb{R}^4$ (Theorem 2.7).
The same argument proves the analogous ground state for $-\Delta u+V(x)u=\lambda u e^{|u|^2}$ in $\mathbb{R}^2$ (Theorem 2.8).
The boundedness and compactness criteria (Theorems 2.1 and 2.2) give a reusable tool for other critical-exponential functionals on $H^2(\mathbb{R}^4)$. For a general nonlinearity, nontrivial radial solutions exist exactly when the potential parameter lies below the Adams ratio $C^*_A$; if the nonlinearity is strong enough at infinity, the threshold is $+\infty$, so solutions exist for every $\gamm

Reading between the lines

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

The strict comparison pattern suggests that the same trapping argument should adapt to periodic or multi-well potentials, where the 'limiting' problems are several copies of the problem at infinity and one expects multibump ground states; the paper does not pursue this.
A testable weakening of the hypotheses is to replace the global bound $V(x)\le\gamma$ by an averaged or localized condition; a direct numerical check with $V(x)=\gamma+\varepsilon(1+|x|^2)^{-1}$ would show whether $m_V<m_\infty$ survives the overshoot.
The Fourier rearrangement step indicates a route to radial minimizers for higher-order or fractional fourth-order problems where pointwise rearrangement fails; whether it works for the fractional Laplacian $(-\Delta)^s$, $s>1$, in $\mathbb{R}^4$ is an open question.

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

Quick read of arXiv:1909.01952. The paper does three things: proves necessary and sufficient conditions for boundedness/compactness of nonlinear functionals in H²(R⁴), proves a ground state for the constant-coefficient biharmonic equation with critical exponential growth, and claims a ground state for a Rabinowitz-type trapping potential. The first two are in good shape; the third is not proven as written.

The compactness criteria (Theorems 2.1 and 2.2) are clean analogues of the known H¹(R²) results, and the proofs look careful. The constant-potential ground state (Theorem 2.6) is a nice application of the Lenzmann–Sok Fourier rearrangement plus the Pohozaev manifold; using that to get a radial minimizing sequence without Pólya–Szegő is genuinely clever. Those parts deserve citation.

The trouble is Theorem 2.7. The proof’s hinge is Lemma 6.2, which claims 0 < m_V < m_∞. The first inequality in that lemma uses ∫V|u|² < ∫γ|u|². That requires V(x) ≤ γ pointwise, but condition (2.2) only says inf V < lim V = γ. V is allowed to exceed γ on bounded sets, in which case the claimed t̃_u > 1 can fail. So the lemma’s assumptions are missing a hypothesis.

The stress-test note is right about a second problem, too. Even if you add V ≤ γ, the paper’s proof builds a sequence by translating the limiting ground state w to infinity. That gives only t_k → 1 and I_V(t_k w(·−y_k)) → m_∞, so the best you get is m_V ≤ m_∞. If V equals γ at infinity, you don’t get the strict gap. And the fixed w, evaluated directly, would give strictness only if V < γ on a set of positive measure—which is not in the stated hypotheses. Strict inequality m_V < m_∞ is used in Lemmas 6.4, 6.6, and 6.13 to rule out vanishing and concentration at infinity, so Theorem 2.7 is unproven as stated. Theorem 2.8 is just a pointer to the same argument, so it inherits the gap.

The good news: the gap is fixable. Add the standard Rabinowitz condition V(x) ≤ γ (and, to get strictness, V < γ on a set of positive measure). Then projecting the fixed limiting ground state w directly—rather than translating it—gives N_V(w) < 0, hence t_w < 1 and I_V(t_w w) < m_∞. The rest of the machinery should go through.

Minor issues: the abstract has a stray parenthesis in the displayed equation, and the Moser sequence in Section 3 has a φ/ψ typo. These are cosmetic.

Verdict: this deserves a serious referee, but not acceptance as is. A skilled PDE person can fix the trapping-potential section, and the compactness criteria are worth keeping. If you work in H²(R⁴) variational problems, cite the first half; don't cite Theorem 2.7 until the proof is repaired.

Referee Report

2 major / 7 minor

Summary. The paper develops variational tools for biharmonic equations with critical exponential growth in R^4. It first proves necessary and sufficient conditions (Theorems 2.1 and 2.2) for boundedness and compactness of nonlinear functionals in H^2(R^4) under Adams-type growth control. These are then used to prove existence of nontrivial radial solutions for (−Δ)^2u+γu=f(u) for γ below an Adams ratio (Theorem 2.3), and, for the special nonlinearity λu exp(2u^2), existence of radial ground states for constant γ when 0<λ<γ (Theorem 2.6), via Fourier rearrangement and a Pohozaev manifold. The main intended novelty is Theorem 2.7, which claims a ground state for the trapping-potential case 0<λ<V0=inf V<lim V=γ<∞, plus an analogous R^2 statement (Theorem 2.8). The constant-potential sections are carefully argued, but the proof of the trapping-potential result has a load-bearing gap in Lemma 6.2, where the strict comparison m_V<m∞ requires a pointwise bound V≤γ that is not part of hypothesis (2.2).

Significance. If the trapping-potential theorem can be repaired, the paper would be a significant contribution: to my knowledge it would be the first ground-state existence result for a biharmonic equation with critical exponential growth and a Rabinowitz-type potential. The compactness criteria in Theorems 2.1–2.2 are useful in their own right, and the use of the Lenzmann–Sok Fourier rearrangement in Section 5 is a clever way to obtain radial minimizing sequences when Schwarz symmetrization is unavailable. The constants 8π², 16π², and 32π² enter through the sharp Adams inequality (1.11), not through curve fitting, and the statements identify explicit thresholds such as the Adams ratio γ*. The main obstacle is the missing pointwise upper bound on V in the proof of Theorem 2.7; this is a correctness risk that can be resolved either by strengthening the hypothesis or by adding a new argument for the strict inequality m_V<m∞.

major comments (2)

[Section 6, Lemma 6.2 (Eqs. (2.2), (6.4), (6.5))] The proof that every u∈N_V has a limiting-Nehari projection \tilde{t}_u>1 uses the strict inequality ∫(|Δu|²+V|u|²)dx < ∫(|Δu|²+γ|u|²)dx, which requires V(x)≤γ pointwise. Hypothesis (2.2) only gives 0<λ<V_0=inf V<lim_{|x|→∞}V=γ; a continuous potential satisfying (2.2) may exceed γ on a bounded set, and for such a u the sign of N∞(u) is not controlled, so \tilde{t}_u may be <1 and estimate (6.5) can fail. This is load-bearing because the strict comparison 0<m_V<m∞ is used in Lemmas 6.4, 6.6, and 6.13 to exclude vanishing and concentration at infinity, and it is the hinge of Theorem 2.7. The authors should either add V(x)≤γ to the hypotheses or prove m_V<m∞ by a different argument. In addition, even under V≤γ, the final line of Lemma 6.2 displays only limsup I_V(u_k)≤m∞; the strict conclusion requires the extra observation that t_k<1 (using V<γ somewhere) and that I∞(tw)<I∞(w) for t<1.
[Section 6, Lemma 6.6] The exclusion of l=0 again depends on the missing bound V≤γ. The proof derives N∞(u_k)=∫(γ−V)|u_k|²+o(1) and then assumes a bounded sequence t_k≥1 with t_k u_k∈N∞, using t_k≥1 in the comparison exp(2t_k²u_k²)≥exp(2u_k²). If V exceeds γ on part of the support, the sign of N∞(u_k) is not controlled and the assertion t_k→1 from (6.11) is not justified. This is a separate appearance of the same hypothesis gap and should be repaired explicitly, even if Lemma 6.2 is fixed.

minor comments (7)

[Abstract and Eq. (0.1)] There is a typo in the displayed equation: it reads “λ s exp(2|s|²))” and should be “λu exp(2|u|²)”, with the extra parenthesis removed.
[Throughout] There are several typos: “Admas” should be “Adams”, “nemely” should be “namely”, “Pólya-Szegö” should have its accents, and “Assumption of V(x)” in Lemma 6.2 should identify the exact hypothesis being used.
[Theorem 2.7] The phrase “non-radial ground state solution” is misleading: since the problem has no symmetry forcing radial solutions, the natural statement is existence of a ground state, which need not be radial. If non-radiality is meant literally, it is not proved.
[Section 6, Lemma 6.13] The step “arguing as Lemma 6.6, we can obtain lim ∫(|Δu''_j|²+V|u''_j|²)dx = ∫(|Δu''_j|²+γ|u''_j|²)dx” is not literally correct because u''_j remains on the right-hand side; it should be stated as lim ∫(γ−V)|u''_j|²dx=0, justified by u''_j→0 in L²_loc and V→γ.
[Section 5, Lemma 5.1] The Fourier rearrangement property used should be stated precisely, including how the Schwarz symmetrization is applied to the complex-valued Fourier transform and which convexity/order property yields the integral inequality for exp(2u²)−1.
[Theorem 2.8] The proof of Theorem 2.8 is not included; if it follows from the same argument, a short explanation of the modifications and of the role of Proposition 1.1 should be added.
[Section 6, Lemma 6.2] The sentence “From Corollary 2.5, we know that m∞ is attained by some w∈N∞” should cite Theorem 2.6 instead, since Corollary 2.5 does not assert ground-state status.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the derivation chain relies on external inequalities and prior theorems, not on fitted inputs or self-referential assumptions.

full rationale

The paper's central claims are Theorem 2.6 (constant potential ground state) and Theorem 2.7 (Rabinowitz-type trapping potential ground state). Theorem 2.6 is built on externally established tools: the sharp Adams inequality with exact growth in R^4 (Masmoudi–Sani, Lu–Tang–Zhu), the Adams inequality (1.9), and the Fourier rearrangement principle of Lenzmann–Sok. The constants 8π^2 and 16π^2 are fixed by these external inequalities, not by fitting or by the target result. Theorem 2.7 uses the standard Nehari-manifold comparison strategy: it introduces the limiting equation with constant potential γ, proves the limiting ground state level m_∞ exists from Theorem 2.6, and then proves m_V < m_∞ via a translated sequence. The comparison m_V < m_∞ is an inequality that must be established; it is not assumed as the conclusion. The paper does not fit any parameter to data and then rename the fit as a prediction. Self-citations to Lam–Lu, Chen–Lu, and Lu–Tang–Zhu appear, but these are citations to published inequalities and extremal results that are independently checkable, not to an unverified premise that already contains the present theorem. The skeptical concern about Lemma 6.2 (that the proof implicitly uses V(x) ≤ γ while assumption (2.2) only gives inf V < γ, and that the translated sequence only yields a non-strict inequality) is a potential correctness gap in the proof, not an instance of circularity. Even if the strict comparison m_V < m_∞ is unjustified, the derivation does not reduce by definition to its own inputs; it is simply an incomplete step. Therefore, no circular step meeting the required evidentiary standard is exhibited, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented physical entities. All constants in the proofs come from known Adams inequalities. The main extra assumption is the unstated pointwise bound V(x) ≤ γ, which is a genuine axiom of the trapping potential argument.

assumptions (5)

standard math Adams' inequality with exact growth in R⁴, equation (1.11), from Masmoudi and Sani.
Used throughout Sections 3 and 5 to bound nonlinear integrals and to establish compactness.
standard math Fourier rearrangement principle from Lenzmann and Sok, cited as [30], used in Lemma 5.1.
Provides the radial minimizing sequence for the Pohozaev manifold in the constant potential case.
standard math Concentration-compactness principle for Adams' inequality on H²(R⁴), invoked in Lemma 6.9.
Used to pass from uniform integrability to equality of local exponential integrals. The principle is cited without proof.
standard math Palais principle of symmetric criticality, invoked to pass from radial critical points to full H²(R⁴) solutions.
Used in Section 4 for Theorem 2.3.
ad hoc to paper V(x) ≤ γ for all x, implicitly used in Lemma 6.2.
The proof of m_V < m_∞ assumes ∫V|u|²dx < ∫γ|u|²dx for functions in the Nehari manifold, which requires V ≤ γ. This is not stated in Theorem 2.7 or in condition (2.2).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials." pith.science (2026). https://pith.science/paper/LYMZBPCE

@misc{pith2026190901952,
  author       = {Pith},
  title        = {Pith review of: Ground states of bi-harmonic equations with critical exponential growth involving constant and trapping potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LYMZBPCE}},
  note         = {Machine review of arXiv:1909.01952}
}

abstract

In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness for a class of nonlinear functionals in $H^{2}(\mathbb{R}^4)$. Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to the semilinear bi-harmonic equation of the form \[ (-\Delta)^{2}u+\gamma u=f(u)\ \text{in}\ \mathbb{R}^4 \] and the range of $\gamma\in \mathbb{R}^{+}$, where $f(s)$ is the general nonlinear term having the critical exponential growth at infinity. Our next goal in this paper is to establish the existence of the ground-state solutions for the equation \begin{equation}\label{con} (-\Delta)^{2}u+V(x)u=\lambda s\exp(2|s|^{2}))\ \text{in}\ \mathbb{R}^{4}, \end{equation} when $V(x)$ is a positive constant using the Fourier rearrangement and the Pohozaev identity. Then we will explore the relationship between the Nehari manifold and the corresponding limiting Nehari manifold to derive the existence of the ground state solutions for the above equation when $V(x)$ is the Rabinowitz type trapping potential, namely it satisfies $$0<V_{0}=\underset{x\in\mathbb{R}^{4}}{\inf}V(x) <\underset{\ | x\ | \rightarrow\infty}{\lim}V(x) < +\infty. $$ The same result and proof applies to the harmonic equation with the critical exponential growth involving the Rabinowitz type trapping potential in $\mathbb{R}^2$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

53 extracted references · 53 canonical work pages

[1]

Adachi, K

S. Adachi, K. Tanaka, Trudinger type inequalities in RN and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057

work page 2000
[2]

Adams, A sharp inequality of J

D.R. Adams, A sharp inequality of J. Moser for higher order deriva tives, Ann. of Math., (2) 128 (1988), 385-398

work page 1988
[3]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Sc. Norm. Sup. Pisa, XVII (1990), 393-413

work page 1990
[4]

F. V. Atkinson, L. A. Peletier, Ground states and Dirichlet proble ms for −∆ u = f (u) in R2, Arch. Rational Mech. Anal., 96 (1986), 147-165

work page 1986
[5]

F. V. Atkinson, L. A. Peletier, Ground states of −∆ u = f (u) and the Emden-Fowler equation, Arch. Rational Mech. Anal., 93 (1986), 103-127

work page 1986
[6]

Alves, M

C. Alves, M. Souto, M. Montenegro, Existence of a ground stat e solution for a nonlinear scalar ﬁeld equation with critical growth, Calc. Var. Partial Diﬀerential Equat ions, 43 (2012), 537–554

work page 2012
[7]

Alves, G

C. Alves, G. Figueiredo, On multiplicity and concentration of positiv e solutions for a class of quasilin- ear problems with critical exponential growth in R2, J. Diﬀerential Equations, 246 (2009), 1288-1311

work page 2009
[8]

Ambrosetti, P

A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in crit ical point theory and applications, J. Funct. Anal., 14 (1973), 349–381

work page 1973

Show all 53 references

[9]

Ambrosetti, M

A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical States of Nonlinear Schrodinger Equations, Arch. Rational Mech. Anal., 140 (1997), 285-300

1997
[10]

Ambrosetti and A

A. Ambrosetti and A. Malchiodi, Perturbation methods and sem ilinear elliptic problems on Rn. Progress in Mathematics, 240. Birkhuser Verlag, Basel, 2006

2006
[11]

Ambrosetti, A

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonliear Schrodinger Equa- tions with potentials, Arch. Rational Mech. Anal., 159 (2001), 253-271

2001
[12]

J. Bao, N. Lam and G. Lu, polyharmonic equations with critical ex ponential growth in the whole space Rn, Discrete and Continuous dynamical Systems, 36 (2016), 577-600

2016
[13]

Brezis and L

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic p roblems involving critical Sovolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477

1983
[14]

L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams inequality a nd ground state solutions to the bi-Laplacian equation in R4, Adv. Nonlinear Stud. 18 (2018), no. 3, 429-452. GROUND STATES OF BI-HARMONIC EQUATIONS WITH CRITICAL EXPON ENTIAL GROWTH 37

2018
[15]

L. Chen, G. Lu and C. Zhang, Sharp weighted TrudingerMoserA dams inequalities on the whole space and the existence of their extremals, Calc. Var. Partial Diﬀe rential Equations, 58(4) (2019), 58:132

2019
[16]

D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var., 3 (1995), 139-153

1995
[17]

Ding and W

W. Ding and W. Ni, On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Rational Mech. Anal. 91 (1986), no. 4, 283-308

1986
[18]

J. M. do ´O, N-Laplacian equations in Rn with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315

1997
[19]

Ibrahim, N

S. Ibrahim, N. Masmoudi and K. Nakanishi, Trudinger-Moser ine quality on the whole plane with the exact growth condition, J. Eur. Math. Soc., 17 (2015), 819-835

2015
[20]

Kang and J

X. Kang and J. Wei, On interacting bumps of semi-classical state s of nonlinear Schrdinger equations, Adv. Diﬀerential Equations 5 (2000), no. 7-9, 899-928

2000
[21]

Kozono, T

H. Kozono, T. Sato and H. Wadade, Upper bound of the best co nstant of a Trudinger-Moser in- equality and its application to a Gagliardo-Nirenberg inequality , Indiana Univ. Math. J., 55 (2006), no. 6, 1951-1974

2006
[22]

N. Lam, G. Lu, Existence of nontrivial solutions to polyharmonic equations with subcritical and critical exponential growth, Discrete and Continuous dynamical S ystems, 32 (2012), 2187-2205

2012
[23]

N. Lam, G. Lu, Existence and multiplicity of solutions to equations of n-Laplacian type with critical exponential growth in Rn, J. Funct. Anal., 262 (2012), 1132-1165

2012
[24]

N. Lam, G. Lu, A new approach to sharp Moser-Trudinger and A dams type inequalities: a rearrangement-free argument, J. Diﬀerential Equations, 255 (2013), 298-325

2013
[25]

Lam and G

N. Lam and G. Lu, Sharp Moser-Trudinger inequality on the Heise nberg group at the critical case and applications, Adv. Math., 231 (2012), 3259-3287

2012
[26]

Lam and G

N. Lam and G. Lu, Sharp Adams type inequalities in Sobolev spaces W m, n m (Rn) for arbitrary integer m, J. Diﬀerential Equations, 253 (2012), 1143-1171

2012
[27]

N. Lam, G. Lu and H. Tang, Sharp subcritical Moser-Trudinger inequalities on Heisenberg groups and subelliptic PDEs, Nonlinear Anal., 95 (2014), 77-92

2014
[28]

N. Lam, G. Lu and L. Zhang, Equivalence of critical and subcritic al sharp Trudinger-Moser-Adams inequalities, Rev. Mat. Iberoam., 33 (2017),1219-1246

2017
[29]

N. Lam, G. Lu and L. Zhang, Sharp singular Trudinger-Moser ine qualities under diﬀerent norms, Advanced Nonlinear Studies, 19 (2) (2019), 2019-2042

2019
[30]

Lenzmann, J

E. Lenzmann, J. Sok, A sharp rearrangement principle in fourie r space and symmetry results for PDEs with arbitrary order, arXiv:1805.06294v1

arXiv
[31]

J. Li, G. Lu and Q. Yang, Fourier analysis and optimal Hardy-Ada ms inequalities on hyperbolic spaces of any even dimension, Adv. Math. 333 (2018), 350-385

2018
[32]

Y. X. Li, B. Ruf, A sharp Trudinger-Moser type inequality for un bounded domains in Rn, Indiana Univ. Math. J., 57 (2008), 451–480

2008
[33]

P. L. Lions, The concentration-compactness principle in the ca lculus of variations. The locally com- pact case, part 1 and 2, Ann. Inst. Henri. Poincar´ e Anal. Non lin´ e aire., 1 (1984), 109-145

1984
[34]

G. Lu, H. Tang, Sharp Moser-Trudinger inequalities on hyperbo lic spaces with exact growth condi- tion, Journal of Geometric Analysis, 26 (2016), 837-857

2016
[35]

G. Lu, H. Tang and M. Zhu, Best Constants for Adams’ Inequa lities with the Exact Growth Condition in Rn, Advanced Nonlinear Studies, 15(4) (2015), 763-788

2015
[36]

Lu and J

G. Lu and J. Wei, On nonlinear Schr¨ odinger equations with totally degenerate potentials. C. R. Acad. Sci. Paris Sr. I Math. 326 (1998), no. 6, 691-696

1998
[37]

Lu and Q

G. Lu and Q. Yang, Sharp Hardy-Adams inequalities for bi-Laplac ian on hyperbolic space of dimen- sion four. Adv. Math. 319 (2017), 567-598

2017
[38]

Masmoudi, F

N. Masmoudi, F. Sani, Adams’ inequality with the exact growth co ndition in R4, Comm. Pure Appl. Math., 67 (2014), 1307-1335. 38 LU CHEN, GUOZHEN LU AND MAOCHUN ZHU

2014
[39]

Masmoudi, F

N. Masmoudi, F. Sani, Trudinger-Moser Inequalities with the Exa ct Growth Condition in R4 and Applications, Comm. Partial Diﬀerential Equations., 40 (2015), 1408-1440

2015
[40]

Masmoudi, F

N. Masmoudi, F. Sani, Higher order Adams’ inequality with the exa ct growth condition, Commun. Contemp. Math., 20 (2018), Doi:10.1142/S0219199717500729

2018 doi
[41]

Moser, Sharp form of an inequality by N

J. Moser, Sharp form of an inequality by N. Trudinger, Indiana U niv. Maths J., 20 (1971), 1077-1092

1971
[42]

Ozawa, On critical cases of Sobolev’s inequalities, J

T. Ozawa, On critical cases of Sobolev’s inequalities, J. Funct. A nal., 127 (1995), 259-269

1995
[43]

Rabinowitz, Minimax methods in critical point theory with applica tions to diﬀerential equations, CBMS Reg

P. Rabinowitz, Minimax methods in critical point theory with applica tions to diﬀerential equations, CBMS Reg. Conf. Ser. Math., 65, American Mathematical Society, P rovidence, RI,1986

1986
[44]

Rabinowitz, On a class of nonlinear Schr¨ oinger equations, Z

P. Rabinowitz, On a class of nonlinear Schr¨ oinger equations, Z. Angew. Math. Phys., 43 (1992), 27-42

1992
[45]

Rabinowitz, Critical point theory and applications to diﬀerent ial equations: a survey, Topological nonlinear analysis, Progr

P. Rabinowitz, Critical point theory and applications to diﬀerent ial equations: a survey, Topological nonlinear analysis, Progr. Nonlinear Diﬀerential Equations Appl., vol. 15, Birkh¨ aser Boston, Boston, MA, 1995, pp. 464–513

1995
[46]

Ruf, A sharp Trudinger-Moser type inequality for unbounde d domains in R2, J

B. Ruf, A sharp Trudinger-Moser type inequality for unbounde d domains in R2, J. Funct. Analysis, 219 (2004), 340-367

2004
[47]

B. Ruf, F. Sani, Ground states for elliptic equations in R2 with exponential critical growth, In: Geometric Properties for Parabolic and Elliptic PDE’s. Springer INdAM Series, 2 (2013). New York: Springer, 251–267

2013
[48]

B. Ruf, F. Sani, Sharp Adams-type inequalities in Rn, Trans. Amer. Math. Soc., 365 (2013), 645-670

2013
[49]

Sani, A biharmonic equation in R4 involving nonlinearities with critical exponential growth, Communications on Pure & Applied Analysis, 12 (2013), 405-428

F. Sani, A biharmonic equation in R4 involving nonlinearities with critical exponential growth, Communications on Pure & Applied Analysis, 12 (2013), 405-428

2013
[50]

N. S. Trudinger, On embeddings in to Orlicz spaces and some applic ations, J. Math. Mech., 17 (1967), 473–484

1967
[51]

Wang, On concentration of positive bound states of nonlinea r Schr¨ odinger equations, Comm

X. Wang, On concentration of positive bound states of nonlinea r Schr¨ odinger equations, Comm. Math. Phys. 153 (1993), no. 2, 229-244

1993
[52]

Yang, Existence of positive solutions to quasilinear equations with exponential growth in the whole Euclidean space, Journal of Functional Analysis, 262 (2012), 1679-1704

Y. Yang, Existence of positive solutions to quasilinear equations with exponential growth in the whole Euclidean space, Journal of Functional Analysis, 262 (2012), 1679-1704

2012
[53]

L. Zhao, Y. Chang, Minimax level estimate for a singular quasilinea r polyharmonic equation in R2m, J. Diﬀerential Equations, 254 (2013), 2434-2464. Lu Chen School of Mathematics and Statistics Beijing Institute of Technology Beijing 100081, P. R. China chenlu5818804@163.com G...

2013

Pith tools

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

[1] [1]

Adachi, K

S. Adachi, K. Tanaka, Trudinger type inequalities in RN and their best exponents, Proc. Amer. Math. Soc., 128 (2000), 2051-2057

work page 2000

[2] [2]

Adams, A sharp inequality of J

D.R. Adams, A sharp inequality of J. Moser for higher order deriva tives, Ann. of Math., (2) 128 (1988), 385-398

work page 1988

[3] [3]

Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Sc. Norm. Sup. Pisa, XVII (1990), 393-413

work page 1990

[4] [4]

F. V. Atkinson, L. A. Peletier, Ground states and Dirichlet proble ms for −∆ u = f (u) in R2, Arch. Rational Mech. Anal., 96 (1986), 147-165

work page 1986

[5] [5]

F. V. Atkinson, L. A. Peletier, Ground states of −∆ u = f (u) and the Emden-Fowler equation, Arch. Rational Mech. Anal., 93 (1986), 103-127

work page 1986

[6] [6]

Alves, M

C. Alves, M. Souto, M. Montenegro, Existence of a ground stat e solution for a nonlinear scalar ﬁeld equation with critical growth, Calc. Var. Partial Diﬀerential Equat ions, 43 (2012), 537–554

work page 2012

[7] [7]

Alves, G

C. Alves, G. Figueiredo, On multiplicity and concentration of positiv e solutions for a class of quasilin- ear problems with critical exponential growth in R2, J. Diﬀerential Equations, 246 (2009), 1288-1311

work page 2009

[8] [8]

Ambrosetti, P

A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in crit ical point theory and applications, J. Funct. Anal., 14 (1973), 349–381

work page 1973