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Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to weighted doubly $D^{1,p}$-critical quasi-linear nonlocal elliptic equations with Hardy potential

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Pith's one-line read This paper proves that every nontrivial nonnegative finite-energy solution of the doubly critical quasi-linear Hartree equation with Hardy potential is radially symmetric and strictly decreasing about the origin, after establishing sharp…

desk verdict The μ>0 extension is substantive, but the claimed near-origin gradient lower bound is false at μ=0; the statement needs to be restricted. read the letter →

arxiv 2502.07816 v1 pith:VTQML32Z submitted 2025-02-09 math.AP

Daomin Cao , Wei Dai , Yafei Li This is my paper · ORCID
classification math.AP MSC 35J9235B0635B40
keywords p-LaplacianHardypotentialHartreenonlinearitydoublycriticalequationradialsymmetrysharpasymptoticestimatesmethodofmovingplanesD^{1p}spaces
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 studies nonnegative finite-energy solutions $u\in D^{1,p}(\mathbb{R}^N)$ (the homogeneous Sobolev space of functions whose gradient lies in $L^p$) of the doubly critical quasi-linear equation $-\Delta_p u - \mu |x|^{-p} u^{p-1} = (|x|^{-2p}\ast u^p) u^{p-1}$ in $\mathbb{R}^N$. Because the Hardy potential is singular at the origin, the equation is not translation invariant, and the authors must first establish sharp two-sided asymptotic estimates for $u$ and $|\nabla u|$ near $0$ and at infinity; these hold for a more general weighted equation with potential $V\in L^{N/(p-s)}(\mathbb{R}^N)$ and weight $|x|^{-s}$. The estimates are governed by two exponents $\gamma_1<\gamma_2$, the roots of $\gamma^{p-2}[(p-1)\gamma^2-(N-p)\gamma]+\mu=0$. With those asymptotics in hand, a refined moving-plane argument shows that every nontrivial nonnegative solution is radially symmetric and strictly decreasing about $0$, and takes the form $u(x)=\lambda^{(N-p)/p}U(\lambda x)$ with $U(\lambda)=1$ and $U'(r)<0$. The result reduces the full classification of solutions to the uniqueness of the radial profile $U$.

What carries the argument

The argument is carried by four interlocking mechanisms. An iteration argument in the standard local regularity theory supplies local integrability, local boundedness and preliminary decay rates for solutions of the generalized equation (1.9); the preliminary rates are then converted into sharp two-sided estimates by scaling. After rescaling $u_R(y)=R^{\gamma_2}u(Ry)$ for large $R$, the sequence converges to a positive solution of the singular model $-\Delta_p u_\infty-\mu|x|^{-p}u_\infty^{p-1}=0$ in $\mathbb{R}^N\setminus\{0\}$, and a classification of such solutions forces $u_\infty(x)=C|x|^{-\gamma_2}$; the same rescaling at the origin produces the lower gradient bounds. The nonlocal term is handled through the exact reflection identity of Lemma 4.1, which rewrites $V_1(x)-V_1(x^\lambda)$ as an integral over the half-space $\Sigma_\lambda$ of $[u^p(y)-u_\lambda^p(y)]$ against a positive kernel, so the moving-plane comparison sees only the sign of $u-u_\lambda$. Finally, strong comparison principles, the weighted Poincaré inequality, and the connectivity of the critical set complete the symmetry proof, with the degenerate case $p\ge2$ and the singular case $1<p<2$ treated separately.

What would settle it

Produce a nontrivial nonnegative $D^{1,p}(\mathbb{R}^N)$-weak solution of (1.1) that is not radially symmetric about $0$, or exhibit a positive solution of the generalized equation (1.9) whose gradient vanishes at a point where the sharp lower bound $|\nabla u|\ge c|x|^{-\gamma-1}$ fails; either observation would refute the main theorem. A more targeted check is to solve the radial equation numerically and measure $\log u(r)/\log r$ near $r=0$ and $r=\infty$: values different from $-\gamma_1$ and $-\gamma_2$ would contradict the sharp asymptotics directly.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.7: for $N\ge 3$, $1<p<N/2$ and $0\le \mu<((N-p)/p)^p$, every nontrivial nonnegative $D^{1,p}(\mathbb{R}^N)$-weak solution of (1.1) lies in $C^{1,\alpha}(\mathbb{R}^N\setminus\{0\})\cap L^\infty_{\mathrm{loc}}(\mathbb{R}^N\setminus\{0\})$ and satisfies the sharp asymptotic estimates $c_0|x|^{-\gamma_1}\le u(x)\le C_0|x|^{-\gamma_1}$ near $0$ and $c_0|x|^{-\gamma_2}\le u(x)\le C_0|x|^{-\gamma_2}$ at infinity, with matching two-sided bounds on $|\nabla u|$; the exponents $\gamma_1<\gamma_2$ are the two roots of $\gamma^{p-2}[(p-1)\gamma^2-(N-p)\gamma]+\mu=0$. Consequently either $u\equiv 0$ or $u>0$ and $u$ is radially symmetric and strictly decreasing about the origin, taking the form $u(x)=\lambda^{(N-p)/p}U(\lambda x)$ with $\lambda=u(1)^{p/(N-p)}$, $U(\lambda)=1$ and $U'(r)<0$ for all $r>0$. The proof is built through the more general weighted equation (1.9) with $0\le s<p$ and $V\in L^{N/(p-s)}(\mathbb{R}^N)$, for which Theorems 1.2–1.4 establish regularity, preliminary decay, and then the sharp asymptotics. Equation (1.1) fits that framework because $V_1(x)=|x|^{-2p}\ast u^p$ is shown to obey the required local integrability, boundedness and decay hypotheses. The same arguments extend to the weighted nonlocal family (1.26) under the extra condition $p_{\sigma,s}\gamma_1+1<N$.

Load-bearing premise

The load-bearing premise is that blow-up limits are classified: the only positive solutions of the singular model $-\Delta_p u-\mu|x|^{-p}u^{p-1}=0$ in $\mathbb{R}^N\setminus\{0\}$ with the observed decay are the fundamental profiles $C|x|^{-\gamma}$ with no critical points; if that classification failed, the sharp gradient bounds and with them the moving-plane proof would collapse.

Editorial extensions

If this is right

  • Every nontrivial nonnegative $D^{1,p}(\mathbb{R}^N)$-weak solution of (1.1) is strictly positive, $C^{1,\alpha}$ away from the origin, and strictly radially decreasing; any non-radial candidate solution is therefore excluded.
  • The sharp asymptotics determine the exact local behavior: $u\sim|x|^{-\gamma_1}$ at $0$ and $u\sim|x|^{-\gamma_2}$ at infinity, with $|\nabla u|$ decaying one power faster, so the estimates (1.20)–(1.23) become the quantitative profile of every solution.
  • When $\mu=0$ the two exponents reduce to $\gamma_1=0$ and $\gamma_2=(N-p)/(p-1)$, recovering the previously known result for the equation without the Hardy potential.
  • For the weighted nonlocal family (1.26), the same sharp estimates and radial symmetry hold whenever $p_{\sigma,s}\gamma_1+1<N$; this extends the result beyond the Hartree endpoint covered by (1.1).
  • The allowed parameter ranges $N\ge3$, $1<p<N/2$ and $0\le\mu<((N-p)/p)^p$ are complete, so the only remaining step toward a full classification is uniqueness of the radial profile $U$.

Reading between the lines

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

  • If uniqueness of the radial profile $U$ is established, Theorem 1.7 immediately yields the full one-parameter family $\lambda^{(N-p)/p}U(\lambda x)$ as the set of all nontrivial nonnegative solutions, mirroring the structure of the classical critical Sobolev equation.
  • The blow-up-and-classify strategy used here suggests an analogous route for fractional or higher-order Hartree equations with a Hardy-type singularity, provided a classification of the limiting singular profile is available.
  • The reflection identity for the nonlocal term indicates that radial symmetry proofs need not rely on reflection positivity; monotonicity of the convolution kernel in half-spaces is enough, which may help in other nonlocal problems where reflection positivity fails.
  • A numerical check is directly testable: solve (1.1) radially near $\mu=((N-p)/p)^p$ and measure $\log u(r)/\log r$; the local exponent should approach $-\gamma_1$ as $r\to0$ and $-\gamma_2$ as $r\to\infty$, and any deviation would signal a missing hypothesis.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies nonnegative D^{1,p}(R^N)-weak solutions of the doubly D^{1,p}-critical quasilinear nonlocal Hartree equation with a Hardy potential, -Δ_p u - μ |x|^{-p} u^{p-1} = (|x|^{-2p} * u^p) u^{p-1} in R^N, for N≥3, 1<p<N/2, 0≤μ<((N-p)/p)^p. The authors first prove regularity and sharp asymptotic estimates near the origin and at infinity for positive solutions of a more general equation (1.9), then use these estimates with the method of moving planes to prove radial symmetry and strict radial monotonicity. A weighted generalization (1.26) is treated in Theorem 1.10. The main theorems include both asymptotic exponents that are explicit roots of an algebraic equation and a complete radial-symmetry statement for all nontrivial nonnegative solutions.

Significance. If the μ>0 part is correct, this is a substantial extension of the nonlocal p-Laplacian classification program from the translation-invariant case μ=0 in [26] to the singular Hardy-potential case, and it provides the nonlocal counterpart of the local results in [77]. The asymptotic exponents are derived explicitly, the μ>0 verification for equation (1.1) is written out in detail, and the moving-plane proof incorporates the necessary weighted Poincaré and critical-set machinery. The paper would be a useful reference for quasilinear nonlocal equations with critical Hardy potentials. However, the claimed endpoint μ=0 in the asymptotic statements is not correct as written, and the moving-plane proof depends on that endpoint, so the central theorem needs revision.

major comments (2)
  1. [Theorem 1.4, Eq. (1.22); Step 3 of §3.3; Remark 1.5; Theorem 1.7] The near-origin gradient lower bound (1.22) is false at μ=0. For μ=0, the characteristic equation has roots γ1=0 and γ2=(N-p)/(p-1), so (1.22) asserts c0|x|^{-1}≤|∇u(x)| for |x|<R0. But the μ=0 case of (1.1) admits the positive radial D^{1,p} solutions of [26]; for any such solution, the radial p-Laplace equation gives r^{N-1}|u'|^{p-2}u' = -∫_0^r s^{N-1}V1(s)u(s)^{p-1}ds = O(r^N) because u and V1 are bounded near 0, hence |u'(r)|=O(r^{1/(p-1)})→0, contradicting c0 r^{-1}. The gap is in Step 3 of Theorem 1.4: the lower bound (3.61) near 0 is stated to be proved 'entirely similar' to the infinity case, but for γ1=0 the rescaled profiles u_R(y)=u(Ry) converge to a bounded p-harmonic function, hence to a constant with identically zero gradient, and the contradiction based on 'the fundamental solution has no critical points' disappears. Consequently (1.22) must be restricted to 0<μ<barμ or replaced by a separate μ=0 statement, and Remark 1.5's assertion that [26, Theorem 1.3] follows from Theorem 1.4 should be corrected. Theorem 1.10, which also allows 0≤μ<barμ, inherits the same problem.
  2. [§4, proof of Theorem 1.7, especially (4.6) and the identification of Zu] The moving-plane argument relies on the singular near-origin estimates precisely at the endpoint μ=0. In particular, (4.6) uses (1.20)-(1.23) to assert sup_{B_{\tilde R0}(0_λ)} u < inf_{B_{\tilde R0}(0_λ)} u_λ for λ<-R1; at μ=0 this inequality is not a consequence of the asymptotics because u is bounded near 0 and u_λ near 0_λ is the reflected value u near 0. Later in Step 2, the bounds (1.22) and (1.23) are used to conclude that the critical set satisfies Z_u ⊂ B_{R1}(0)\overline{B_{R0}(0)}; at μ=0 the origin is a critical point for the radial solutions of [26], so this conclusion is false. Thus the proof of Theorem 1.7 as written does not cover μ=0. This is fixable by stating the theorem for 0<μ<barμ and invoking [26] separately at μ=0, or by developing a distinct near-origin analysis for γ1=0.
minor comments (4)
  1. [Theorem 1.10 statement] Theorem 1.10 states u ∈ C^{1,α}(R^N\setminus\{0\}) ∩ L^∞(R^N\setminus\{0\}); for μ>0 the solution satisfies u ~ |x|^{-γ1} near 0, so it is not globally bounded away from 0. The intended space is L^∞_loc(R^N\setminus\{0\}), as in Theorem 1.7.
  2. [Theorem 1.10 statement] The phrase 'u > 0 in R^N' in Theorem 1.10 should be 'u > 0 in R^N\setminus\{0\}', since for μ>0 the solution is singular at the origin and is not defined there in the classical sense.
  3. [Remark 1.11] The equivalence asserted in Remark 1.11 between the condition on σ+s, p_{s,σ}, μ and the inequality p_{s,σ}γ1+1<N is stated without derivation; a short proof or reference for this algebraic equivalence would improve readability.
  4. [Remark 1.5] Remark 1.5 says that Theorem 1.4 reduces to [26, Theorem 1.3] when μ=0. As explained in the first major comment, the near-origin gradient lower bound in (1.22) is incompatible with the bounded radial solutions classified in [26], so this remark should be reworded after the endpoint issue is resolved.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is self-contained, and the only overlapping-author citation is not load-bearing.

full rationale

The derivation chain is genuinely self-contained. The sharp asymptotic exponents gamma1 and gamma2 are obtained as the two roots of the explicit characteristic equation gamma^{p-2}[(p-1)gamma^2-(N-p)gamma]+mu=0, so they are not fitted or imported from the conclusion. The load-bearing external inputs are the classification results [43, Theorems 1.3 and 4.1], the asymptotic comparison lemmas from [95] and [96], and the moving-plane machinery from [77]; none of these works shares authors with the present paper, so no uniqueness theorem is being imported from the authors' own prior work. The only citation with overlapping authors is [26] (Dai, Li and Liu), and it is used only as the mu=0 baseline that the paper extends and recovers in Remark 1.5, not as a premise on which the mu>0 proof depends. For the nonlocal equation (1.1), the verification that V1(x)=|x|^{-2p}*u^p satisfies the hypotheses of Theorem 1.4 is done through the preliminary estimates of Theorem 1.3 and Lemma 3.9, not through the sharp estimates being proved, so there is no bootstrap circularity. The reviewer's concern that the mu=0 gradient lower bound (1.22) may be false is a mathematical correctness issue about the omitted 'entirely similar' Step 3 argument, not a circularity of the paper's reasoning, and therefore is not scored in this pass.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central proof rests on standard inequalities, regularity theory, comparison principles, and one external classification theorem. The paper introduces no new physical entities or fitted numerical parameters; the exponents gamma1, gamma2 and tau1, tau2 are determined by the equation and the proof, not by data fitting. The main load-bearing external input is the classification of limiting profiles from [43], which the paper cites but does not derive.

assumptions (9)
  • standard math Hardy-Littlewood-Sobolev inequality and double weighted Hardy-Littlewood-Sobolev inequality hold with the stated exponents.
    Used throughout Section 3 to estimate the convolution terms V1 and V3; stated as Theorems 2.1 and 2.2 and cited to [46,47,60,67,86,17].
  • standard math Caffarelli-Kohn-Nirenberg and Hardy-Sobolev inequalities hold in D^{1,p}(R^N).
    Stated as Theorem 2.3 and Remark 2.4; used in the proof of Lemma 3.4 and in the I4 estimates of Section 4.
  • standard math Sobolev embedding D^{1,p}(R^N) into L^{p*}(R^N) with p*=Np/(N-p) holds.
    Invoked repeatedly, for example in (1.2), (1.3) and in Lemma 3.1.
  • standard math C^{1,alpha} local regularity estimates hold for weak solutions of p-Laplace equations with the stated source terms.
    Used to obtain u in C^{1,alpha}(R^N\{0}); cited to DiBenedetto [39], Tolksdorf [90], and Kuusi-Mingione [55].
  • standard math Strong maximum principle and Hopf's lemma hold for the p-Laplacian in the forms stated in Lemma 2.12.
    Used to conclude either u identically 0 or u>0 in R^N\{0} and to prove strict radial monotonicity at the end of Theorem 1.7.
  • standard math Strong comparison principles for p-Laplace equations hold in the forms stated in Lemmas 2.10 and 2.11.
    Used in Step 2 of the moving-plane proof to compare u and u_lambda in half-spaces; cited to [37] and [30].
  • domain assumption The classification theorem from [43, Theorems 1.3 and 4.1] for positive solutions of -Delta_p u - mu |x|^{-p} u^{p-1}=0 in R^N\{0} under sharp asymptotic conditions is valid.
    This external result supplies the limiting profile u_infinity = C |x|^{-gamma2} in the contradiction proof of Theorem 1.4 Step 3. It is load-bearing for the lower gradient bounds.
  • standard math Weighted Poincare type inequalities hold for the weight rho=|grad u|^{p-2} with p>=2, as stated in Lemma 2.18 and Remark 2.19.
    Used in Section 4 to control the integral over S^lambda_delta via Lemma 2.18; cited to [36,45,56] and justified by the gradient integrability estimates.
  • domain assumption The extremal/minimizer existence results of Su and Chen [87] provide a nontrivial nonnegative solution to (1.1) as the Euler-Lagrange equation of (1.6).
    Used in the introduction to motivate that the solution class is nonempty; it does not affect the symmetry proof itself.

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Pith. "Pith review of Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to weighted doubly $D^{1,p}$-critical quasi-linear nonlocal elliptic equations with Hardy potential." pith.science (2026). https://pith.science/paper/VTQML32Z

@misc{pith2026250207816,
  author       = {Pith},
  title        = {Pith review of: Radial symmetry and sharp asymptotic behaviors of nonnegative solutions to weighted doubly $D^1,p$-critical quasi-linear nonlocal elliptic equations with Hardy potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTQML32Z}},
  note         = {Machine review of arXiv:2502.07816}
}
abstract

In this paper, we mainly consider nonnegative weak solutions $u\in D^{1,p}(\R^{N})$ to the doubly $D^{1,p}(\R^{N})$-critical nonlocal quasi-linear Schr\"{o}dinger-Hartree equation: \begin{align*} -\Delta_p u- \mu \frac{u^{p-1}}{|x|^p}=\left(|x|^{-2p}\ast |u|^{p}\right)|u|^{p-2}u \qquad &\mbox{in} \,\, \mathbb{R}^N, \end{align*} where $N\geq3$, $0\leq\mu< \bar{\mu}:=\left( (N-p)/p \right)^p$ and $1<p<\frac{N}{2}$. When $\mu>0$, due to appearance of the Hardy potential, the equation has singularity at $0\in\mathbb{R}^{N}$ and hence is not translation invariant, so sharp asymptotic estimates near the origin must be involved. First, we establish regularity and the sharp estimates on asymptotic behaviors near the origin and the infinity for any positive solution $u\in D^{1,p}(\R^{N})$ (and $|\nabla u|$) to more general equation $-\triangle_p u - \mu \frac{1}{|x|^p}u^{p-1}=V(x)\frac{1}{|x|^s}u^{p-1}$ with $N\geq2$, $0\leq\mu< \bar{\mu}$, $1<p<N$, $0\leq s < p$ and $0\leq V(x)\in L^\frac{N}{p-s}(\R^N)$. Then, as a consequence, we can apply the method of moving planes to prove that all the nontrivial nonnegative solutions in $D^{1,p}(\R^{N})$ are radially symmetric and strictly radially decreasing about the origin $0\in\mathbb{R}^{N}$. The sharp asymptotic estimates and radial symmetry for more general weighted doubly $D^{1,p}$-critical nonlocal quasi-linear equations were also derived. Our results extend the results in \cite{DLL} from the special case $\mu=0$ to general cases $0\leq\mu<\bar{\mu}$.

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Works this paper leans on

98 extracted references · 76 canonical work pages · cited by 1 Pith paper

  1. [26]

    W. Dai, Y. Li and Z. Liu. Radial symmetry and sharp asymptotic behaviors of nonnegat ive solutions to D1,p - critical quasi-linear static Schr¨ odinger-Hartree equation involving p-Laplacian −∆ p, Math. Ann., 391 (2024), no. 2, 2653–2708. 56 DAOMIN CAO, WEI DAI, YAFEI LI

    work page 2024

  2. [77]

    Oliva, B

    F. Oliva, B. Sciunzi and G. Vaira, Radial symmetry for a quasilinear elliptic equation with a c ritical Sobolev growth and Hardy potential , J. Math. Pures Appl., 140 (2020), 89–109

    work page 2020

  3. [1]

    C. O. Alves, Existence of positive solutions for a problem with lack of co mpactness involving the p-Laplacian, Nonlinear Anal., 51 (2002), no. 7, 1187–1206

    2002

  4. [2]

    C. O. Alves and M. Yang, Investigating the multiplicity and concentration behavio ur of solutions for a quas- linear Choquard equation via the penalization method , Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), no. 1, 23–58

    2016

  5. [3]

    M. F. Bidaut-V´ eron, Local and global behavior of solutions of quasilinear equat ions of Emden-Fowler type , Arch. Ration. Mech. Anal., 107 (1989), no. 4, 293–324

    1989

  6. [4]

    Caffarelli, B

    L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear ellip tic equations with critical Sobolev growth , Comm. Pure Appl. Math., 42 (1989), no. 3, 271–297

    1989

  7. [5]

    Caffarelli, R

    L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequalities with weights , Compositio Math., 53 (1984), no. 3, 259–275

    1984

  8. [6]

    Cao and W

    D. Cao and W. Dai, Classification of nonnegative solutions to a bi-harmonic eq uation with Hartree type non- linearity, Proc. Royal Soc. Edinburgh-A: Math., 149 (2019), no.4, 979–994

    2019

Show all 98 references
  1. [7]

    D. Cao, W. Dai and Y. Zhang, Existence and symmetry of solutions to 2-D Schr¨ odinger-Ne wton equations , Dyn. Partial Differ. Equ., 18 (2021), no. 2, 113–156

    2021

  2. [8]

    Cao and P

    D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential , J. Differential Equations, 205 (2004), no. 2, 521–537

    2004

  3. [9]

    Cao and S

    D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problems w ith critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), no. 2, 424–434

    2003

  4. [10]

    D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth , Journal of Functional Analysis, 2012, 262(6): 2861–2902

    2012

  5. [11]

    Cao and S

    D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var. Partial Differential Equations, 38 (2010), no. 3-4, 471–501

    2010

  6. [12]

    Catino, D

    G. Catino, D. D. Monticelli and A. Roncoroni, On the critical p-Laplace equation, Adv. Math., 433 (2023), Paper No. 109331, 38 pp

    2023

  7. [13]

    Catrina and Z

    F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp cons tants, existence (and nonexistence), and symmetry of extremal functions , Comm. Pure Appl. Math., 54 (2001), no. 2, 229–258

    2001

  8. [14]

    W. Chen, W. Dai and G. Qin, Liouville type theorems, a priori estimates and existence o f solutions for critical and super-critical order Hardy-H´ enon type equations in Rn, Math. Z., 303 (2023), Paper No. 104, 36 pp

    2023

  9. [15]

    Chen and C

    W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equa tions, Duke Math. J., 62 (1991), no. 3, 615–622

    1991

  10. [16]

    Chen and C

    W. Chen and C. Li, A priori estimates for prescribing scalar curvature equati ons, Ann. of Math., 145 (1997), no. 3, 547–564

    1997

  11. [17]

    Chen and C

    W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev i nequality, Proc. Amer. Math. Soc., 136 (2008), no.3, 955–962

    2008

  12. [18]

    W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacia n, Adv. Math., 308 (2017), 404–437

    2017

  13. [19]

    W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation , Comm. Pure Appl. Math., 59 (2006), no. 3, 330–343

    2006

  14. [20]

    Chen and Y

    L. Chen and Y. Yang, Classification of positive solutions of Hardy-Sobolev equa tion without the finite volume constraints, 2023, preprint, arXiv:2312.16017, 21 pp

    2023

  15. [21]

    Cingolani and G

    S. Cingolani and G. Vannella, Multiple positive solutions for a critical quasilinear equ ation via Morse theory , Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire,26 (2009), no. 2, 397–413

    2009

  16. [22]

    Ciraolo, A

    G. Ciraolo, A. Figalli and A. Roncoroni, Symmetry results for critical anisotropic p-Laplacian equations in convex cones, Geom. Funct. Anal., 30 (2020), no. 3, 770–803

    2020

  17. [23]

    W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang, Regularity and classification of solutions to static Hartre e equations involving fractional Laplacians , Discrete and Continuous Dynamical Systems - A, 39 (2019), no. 3, 1389–1403

    2019

  18. [24]

    W. Dai, Y. Fang and G. Qin, Classification of positive solutions to fractional order Ha rtree equations via a direct method of moving planes , J. Differential Equations, 265 (2018), 2044–2063

    2018

  19. [25]

    W. Dai, C. Gui and Y. Luo, Anisotropic Finsler N -Laplacian Liouville equation in convex cones , 2024, preprint, submitted for publication, arXiv:2407.04987, 38 pp

    2024 arXiv

  20. [27]

    Dai and Z

    W. Dai and Z. Liu, Classification of nonnegative solutions to static Schr¨ odi nger-Hartree and Schr¨ odinger- Maxwell equations with combined nonlinearities , Calc. Var. Partial Differential Equations, 58 (2019), no. 4, Paper No. 156, 24 pp

    2019

  21. [28]

    W. Dai, Z. Liu and G. Qin, Classification of nonnegative solutions to static Schr¨ odi nger-Hartree-Maxwell type equations, SIAM J. Math. Anal., 53 (2021), no. 2, 1379–1410

    2021

  22. [29]

    Dai and G

    W. Dai and G. Qin, Classification of nonnegative classical solutions to third -order equations, Adv. Math., 328 (2018), 822–857

    2018

  23. [30]

    Damascelli, Comparison theorems for some quasilinear degenerate ellip tic operators and applications to symmetry and monotonicity results , Ann

    L. Damascelli, Comparison theorems for some quasilinear degenerate ellip tic operators and applications to symmetry and monotonicity results , Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire.,15 (1998), no. 4, 493–516

    1998

  24. [31]

    Damascelli, A

    L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden- Fowler type, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire,26 (2009), no. 4, 1099–1119

    2009

  25. [32]

    Damascelli, S

    L. Damascelli, S. Merch´ an, L. Montoro and B. Sciunzi, Radial symmetry and applications for a problem in- volving the −∆ p(·) operator and critical nonlinearity in RN , Advances in Mathematics, 265 (2014), 313–335

    2014

  26. [33]

    Damascelli and F

    L. Damascelli and F. Pacella, Monotonicity and symmetry of solutions of p-Laplace equations via the moving plane method , Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), no. 4, 689–707

    1998

  27. [34]

    Damascelli, F

    L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of p-Laplace equations via the moving plane method , Arch. Ration. Mech. Anal., 148 (1999), no. 4, 291–308

    1999

  28. [35]

    Damascelli and M

    L. Damascelli and M. Ramaswamy, Symmetry of C1 solutions of p-Laplace equations in RN , Adv. Nonlinear Stud., 1 (2001), no. 1, 40–64

    2001

  29. [36]

    Damascelli and B

    L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutio ns of m-Laplace equa- tions, J. Differential Equations, 206 (2004), no. 2, 483–515

    2004

  30. [37]

    Damascelli and B

    L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differ. Equ., 25 (2006), no. 2, 139–159

    2006

  31. [38]

    E. N. Dancer, H. Yang and W. Zou, Liouville-type results for a class of quasilinear elliptic systems and appli- cations. (English summary) , J. Lond. Math. Soc., 99 (2019), no. 2, 273–294

    2019

  32. [39]

    DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic e quations, Nonlinear Anal., 7 (1983), no

    E. DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic e quations, Nonlinear Anal., 7 (1983), no. 8, 827–850

    1983

  33. [40]

    Dipierro, Geometric inequalities and symmetry results for elliptic s ystems, Discrete Contin

    S. Dipierro, Geometric inequalities and symmetry results for elliptic s ystems, Discrete Contin. Dyn. Syst.-A, 33 (2013), no. 8, 3473–3496

    2013

  34. [41]

    Dipierro, L

    S. Dipierro, L. Montoro, I. Peral and B. Sciunzi, Qualitative properties of positive solutions to nonlocal c ritical problems involving the Hardy-Leray potential , Calc. Var. Partial Differential Equations, 55 (2016), no. 4, Paper No. 99, 29 pp

    2016

  35. [42]

    H. Dong, F. Peng, Y. R.-Y. Zhang and Y. Zhou, Hessian estimates for equations involving p-Laplacian via a fundamental inequality , Adv. Math., 370 (2020), Paper No. 107212, 40 pp

    2020

  36. [43]

    Esposito, L

    F. Esposito, L. Montoro, B. Sciunzi and D. Vuono, Asymptotic behaviour of solutions to the anisotropic doubl y critical equation, Calc. Var. Partial Differential Equations, 63 (2024), no.3, Paper No. 77, 44 pp

    2024

  37. [44]

    Farina, B

    A. Farina, B. Sciunzi and E. Valdinoci, On a Poincar´ e type formula for solutions of singular and deg enerate elliptic equations , Manuscripta Math., 132 (2010), no. 3-4, 335–342

    2010

  38. [45]

    Ferrari and E

    F. Ferrari and E. Valdinoci, Some weighted Poincar´ e inequalities, Indiana Univ. Math. J., 58 (2009), no. 4, 1619–1637

    2009

  39. [46]

    R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobo lev inequality , Calc. Var. & Partial Differential Equations, 39 (2010), 85–99

    2010

  40. [47]

    R. L. Frank and E. H. Lieb, A new, rearrangement-free proof of the sharp Hardy-Littlew ood-Sobolev inequality. Spectral theory, function spaces and inequalities , Oper. Theory Adv. Appl., Birkh¨ auser/Springer Basel AG, Basel, 219 (2012), 55–67

    2012

  41. [48]

    Frohlich and E

    J. Frohlich and E. Lenzmann, Mean-field limit of quantum bose gases and nonlinear Hartree equation, in: Sminaire E. D. P. (2003-2004), Expos nXVIII. 26p

    2003

  42. [49]

    Gidas, W

    B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle , Comm. Math. Phys., 68 (1979), no. 3, 209–243

    1979

  43. [50]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order , 2nd Edition, Springer, Berlin, 1983

    1983

  44. [51]

    Guedda and L

    M. Guedda and L. Veron, Local and global properties of solutions of quasilinear ell iptic equations, J. Differential Equations, 76 (1988), no. 1, 159–189

    1988

  45. [52]

    L. Guo, T. Hu, S. Peng and W. Shuai, Existence and uniqueness of solutions for Choquard equatio n involving Hardy-Littlewood-Sobolev critical exponent , Calc. Var. Partial Differential Equations, 58 (2019), no. 4, Paper No. 128, 34 pp. NONLOCAL p-LAPLACE EQUATIONS WITH HARDY PO...

    2019

  46. [53]

    Guo and J

    Y. Guo and J. Liu, Solutions of p-sublinear p-Laplacian equation via Morse theory. , J. London Math. Soc., 72 (2005), no. 3, 632–644

    2005

  47. [54]

    Han and F

    Q. Han and F. Lin, Elliptic partial differential equations. Second edition. C ourant Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011. x+147 pp. ISBN: 978-0-8218-5313-9

    2011

  48. [55]

    Kuusi and G

    T. Kuusi and G. Mingione, Universal potential estimates , J. Funct. Anal., 262 (2012), no.10, 4205–4269

    2012

  49. [56]

    Le and D

    P. Le and D. H. T. Le, Classification of positive solutions to p-Laplace equations with critical Hardy-Sobolev exponent, Nonlinear Analysis: Real World Applications, 74 (2023), Paper No. 103949, 13 pp

    2023

  50. [57]

    Lei, Qualitative analysis for the static Hartree-type equation s, SIAM J

    Y. Lei, Qualitative analysis for the static Hartree-type equation s, SIAM J. Math. Anal., 45 (2013), no. 1, 388–406

    2013

  51. [58]

    D. Li, C. Miao and X. Zhang, The focusing energy-critical Hartree equation , J. Diff. Equations, 246 (2009), 1139–1163

    2009

  52. [59]

    E. H. Lieb, Existence and uniqueness of the minimizing solution of Choq uard’s nonlinear equation, Studies in Appl. Math., 57 (1977), no. 2, 93–105

    1977

  53. [60]

    E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and relat ed inequalities, Ann. of Math., 118 (1983), 349–374

    1983

  54. [61]

    E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems , Comm. Math. Phys., 53 (1977), 185–194

    1977

  55. [62]

    G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic e quations, Nonlinear Anal., 12 (1988), no. 11, 1203–1219

    1988

  56. [63]

    C. S. Lin, A classification of solutions of a conformally invariant fou rth order equation in Rn, Comment. Math. Helv., 73 (1998), 206–231

    1998

  57. [64]

    Lin and X

    D. Lin and X. Ma, Best constant and extremal functions for a class Hardy-Sobo lev-Maz’ya inequalities, preprint, arXiv: 2412.09033

    arXiv

  58. [65]

    P. L. Lions, The Choquard equation and related questions , Nonlinear Anal., 4 (1980), 1063–1072

    1980

  59. [66]

    P. L. Lions, The concentration-compactness principle in the calculus o f variations , Rev. Mat. Iberoam., 1 (1985), no. 1, 145–201

    1985

  60. [67]

    P. L. Lions, The concentration-compactness principle in the calculus o f variations. The limit case, part 2 , Rev. Mat. Iberoam., 1 (1985), no. 2, 45–121

    1985

  61. [68]

    Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796–1806

    S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796–1806

    2009

  62. [69]

    Ma and L

    L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlin ear Choquard equation , Arch. Rational Mech. Anal., 195 (2010), no. 2, 455–467

    2010

  63. [70]

    Ma and Q

    X. Ma and Q. Ou, A Liouville theorem for a class semilinear elliptic equatio ns on the Heisenberg group , Adv. Math., 413 (2023), Paper No. 108851, 20 pp

    2023

  64. [71]

    X. Ma, Q. Ou and T. Wu, Jerison-Lee identities and Semi-linear subelliptic equat ions on CR manifolds , preprint, arXiv:2311.16428, 38 pp

    arXiv

  65. [72]

    Merch´ an, L

    S. Merch´ an, L. Montoro, I. Peral and B. Sciunzi,Existence and qualitative properties of solutions to a quas ilinear elliptic equation involving the Hardy-Leray potential , Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire,31 (2014), no.1, 1–22

    2014

  66. [73]

    C. Miao, G. Xu and L. Zhao, Global well-posedness, scattering and blow-up for the ener gy-critical, focusing Hartree equation in the radial case , Colloq. Math., 114 (2009), 213–236

    2009

  67. [74]

    Moroz and J

    V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative prop- erties and decay asymptotics , J. Funct. Anal., 265 (2013), no. 2, 153–184

    2013

  68. [75]

    Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations , Comm

    J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations , Comm. Pure Appl. Math., 13 (1960), 457–468

    1960

  69. [76]

    H. M. Nguyen and M. Squassina, On Hardy and Caffarelli-Kohn-Nirenberg inequalities , J. Anal. Math. 139 (2019), no.2, 773–797

    2019

  70. [78]

    Ou, On the classification of entire solutions to the critical p-Laplace equation, preprint, arXiv: 2210.05141

    Q. Ou, On the classification of entire solutions to the critical p-Laplace equation, preprint, arXiv: 2210.05141

    arXiv

  71. [79]

    Pol´ aˇ cik, P

    P. Pol´ aˇ cik, P. Quittner and P. Souplet,Singularity and decay estimates in superlinear problems vi a Liouville- type theorems. I. Elliptic equations and systems , Duke Math. J., 139 (2007), no. 3, 555–579

    2007

  72. [80]

    Pucci and J

    P. Pucci and J. Serrin, The Maximum Principle , Birkh¨ auser, Boston, 2007

    2007

  73. [81]

    Sciunzi, Classification of positive D1,p (RN )-solutions to the critical p-Laplace equation in RN , Advances in Mathematics, 291 (2016), 12–23

    B. Sciunzi, Classification of positive D1,p (RN )-solutions to the critical p-Laplace equation in RN , Advances in Mathematics, 291 (2016), 12–23. 58 DAOMIN CAO, WEI DAI, YAFEI LI

    2016

  74. [82]

    Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term , Commu- nications in Contemporary Mathematics, 16 (2014), no

    B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term , Commu- nications in Contemporary Mathematics, 16 (2014), no. 6, Paper No. 1450013

    2014

  75. [83]

    Serrin, Local behavior of solutions of quasi-linear equations , Acta Math., 111 (1964), 247–302

    J. Serrin, Local behavior of solutions of quasi-linear equations , Acta Math., 111 (1964), 247–302

    1964

  76. [84]

    Serrin, A symmetry problem in potential theory , Arch

    J. Serrin, A symmetry problem in potential theory , Arch. Ration. Mech. Anal., 43 (1971), no. 4, 304–318

    1971

  77. [85]

    Serrin and H

    J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for q uasilinear elliptic equations and inequalities, Acta Math., 189 (2002), no. 1, 79–142

    2002

  78. [86]

    E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space , Indiana Univ. Math. J., 7 (1958), no. 4, 503–514

    1958

  79. [87]

    Su and H

    Y. Su and H. Chen, The minimizing problem involving p-Laplacian and Hardy-Li ttlewood-Sobolev upper critical exponent, Electron. J. Qual. Theory Differ. Equ., 2018, Paper No. 74, 16 pp

    2018

  80. [88]

    Talenti, Best constant in Sobolev inequality , Ann

    G. Talenti, Best constant in Sobolev inequality , Ann. Mat. Pura Appl., 110 (1976), no. 4, 353–372

    1976

  81. [89]

    Teixeira, Regularity for quasilinear equations on degenerate singul ar sets , Math

    E. Teixeira, Regularity for quasilinear equations on degenerate singul ar sets , Math. Ann., 358 (2014), no. 1-2, 241–256

    2014

  82. [90]

    Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J

    P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), no. 1, 126–150

    1984

  83. [91]

    N. S. Trudinger, Remarks concerning the conformal deformation of Riemannia n structures on compact mani- folds, Ann. Sc. Norm. Super. Pisa, 22 (1968), no. 3, 265–274

    1968

  84. [92]

    J. L. V´ azquez, A strong maximum principle for some quasilinear elliptic eq uations, Appl. Math. Optim., 12 (1984), no. 3, 191–202

    1984

  85. [93]

    V´ etois, A priori estimates and application to the symmetry of soluti ons for critical p–Laplace equations, J

    J. V´ etois, A priori estimates and application to the symmetry of soluti ons for critical p–Laplace equations, J. Differential Equations, 260 (2016), no. 1, 149–161

    2016

  86. [94]

    Wei and X

    J. Wei and X. Xu, Classification of solutions of higher order conformally inv ariant equations, Math. Ann., 313 (1999), no. 2, 207–228

    1999

  87. [95]

    C. L. Xiang, Asymptotic behaviors of solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential, Journal of Differential Equations, 259 (2015), no. 8, 3929–3954

    2015

  88. [96]

    C. L. Xiang, Gradient estimates for solutions to quasilinear elliptic e quations with critical Sobolev growth and hardy potential, Acta Math. Sci. Ser. B (Engl. Ed.), 37 (2017), no.1, 58–68

    2017

  89. [97]

    Zhang and S

    Z. Zhang and S. Li, On sign-changing and multiple solutions of the p-Laplacian, J. Funct. Anal. 197 (2003), no. 2, 447–468

    2003

  90. [98]

    Zhou, Classification theorem for positive critical points of Sobo lev trace inequality , 2024, preprint, arXiv:2402.17602, 48 pp

    Y. Zhou, Classification theorem for positive critical points of Sobo lev trace inequality , 2024, preprint, arXiv:2402.17602, 48 pp. Institute of Applied Mathematics, Chinese Academy of Scien ces, Beijing 100190, and University of Chinese Academy of Sciences, Beijing 100049, P....

    2024 arXiv

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