ELLIPTIC EQUATION WITH CRITICAL EXPONENT AND DIPOLE POTENTIAL: EXISTENCE AND DECAY ESTIMATES

  • Yu SU ,
  • Zhisu LIU ,
  • Senli LIU
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  • 1. School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, China;
    2. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China;
    3. Institute for Advanced Marine Research, China University of Geosciences, Guangzhou 511462, China;
    4. College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China
Yu Su, E-mail: yusumath@aust.edu.cn;Senli Liu, E-mail: liusl@hnust.edu.cn

Received date: 2023-06-07

  Revised date: 2024-04-11

  Online published: 2025-05-08

Supported by

Yu Su's research was supported by the Natural Science Research Project of Anhui Educational Committee(2023AH040155). Zhisu Liu's research was supported by the Guangdong Basic and Applied Basic Research Foundation (2023A1515011679; 2024A1515012704) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (CUG2106211; CUGST2).

Abstract

The paper is concerned with a class of elliptic equation with critical exponent and Dipole potential. More precisely, we make use of the refined Sobolev inequality with Morrey norm to obtain the existence and decay properties of nonnegative radial ground state solutions.

Cite this article

Yu SU , Zhisu LIU , Senli LIU . ELLIPTIC EQUATION WITH CRITICAL EXPONENT AND DIPOLE POTENTIAL: EXISTENCE AND DECAY ESTIMATES[J]. Acta mathematica scientia, Series B, 2025 , 45(2) : 636 -658 . DOI: 10.1007/s10473-025-0220-9

References

[1] Ambrosio L, Prato G, Mennucci A.Introduction to Measure Theory and Integration. Pisa: Edizioni della Normale, 2011
[2] Amadori A, Gladiali F. Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem. Calc Var Partial Differential Equations, 2019, 58: Art 168
[3] Aubin T. Problèmes isopérimétriques et espaces de Sobolev. J Differential Geometry, 1976, 11: 573-598
[4] Barrios B, Quaas A. The sharp exponent in the study of the nonlocal Hénon equation in $\mathbb{R}^{N}$: a Liouville theorem and an existence result. Calc Var Partial Differential Equations, 2020, 59: Art 114
[5] Bonheure D, Casteras J, Gladiali F. Bifurcation analysis of the Hardy-Sobolev equation. J Differential Equations, 2021, 296: 759-798
[6] Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc, 1983, 88: 486-490
[7] Berestycki H, Lions P. Nonlinear scalar field equations, I: Existence of a ground state. Arch Rational Mech Anal, 1983, 82: 313-345
[8] Cao D, Han P. Solutions to critical elliptic equations with multi-singular inverse square potentials. J Differential Equations, 2006, 224: 332-372
[9] Cao D, Peng S. Asymptotic behavior for elliptic problems with singular coefficient and nearly critical Sobolev growth. Ann Mat Pura Appl, 2006, 185: 189-205
[10] Chou K, Chu C. On the best constant for a weighted Sobolev-Hardy inequality. J London Math Soc, 1993, 48: 137-151
[11] Fall M, Felli V. Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential. J Funct Anal, 2014, 267: 1851-1877
[12] Fall M, Felli V. Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete Contin Dyn Syst, 2015, 35: 5827-5867
[13] Fanelli L, Felli V, Veronica M, Primo A. Time decay of scaling invariant electromagnetic Schrödinger equations on the plane. Comm Math Phys, 2015, 337: 1515-1533
[14] Felli V, Terracini S. Nonlinear Schrödinger equations with symmetric multi-polar potentials. Calc Var Partial Differential Equations, 2006, 27: 25-58
[15] Felli V, Terracini S. Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Comm Partial Differential Equations, 2006, 31: 469-495
[16] Felli V, Marchini E, Terracini S. On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity. Discrete Contin Dyn Syst, 2008, 21: 91-119
[17] Felli V, Ferrero A, Terracini S. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete Contin Dyn Syst, 2012, 32: 3895-3956
[18] Felli V, Ferrero A. Almgren-type monotonicity methods for the classification of behaviour at corners of solutions to semilinear elliptic equations. Proc Roy Soc Edinburgh Sect A, 2013, 143: 957-1019
[19] Felli V, Ferrero A, Terracini S. Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J Eur Math Soc (JEMS), 2011, 13: 119-174
[20] Filippucci R, Pucci P, Robert F. On a $p$-Laplace equation with multiple critical nonlinearities. J Math Pures Appl, 2009, 91: 156-177
[21] Ni W. A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ Math J, 1982, 31: 801-807
[22] Ghoussoub N, Yuan C. Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Amer Math Soc, 2000, 352: 5703-5743
[23] Gidas B, Spruck J. Global and local behavior of positive solutions of nonlinear elliptic equations. Commun Pure Appl Math, 1981, 34: 525-598
[24] Gladiali F, Grossi M, Neves S. Nonradial solutions for the Hénon equation in $\mathbb{R}^{N}$. Adv Math, 2013, 249: 1-36
[25] Han P, Liu Z. Solutions to nonlinear Neumann problems with an inverse square potential. Calc Var Partial Differential Equations, 2007, 30: 315-352
[26] Kang D, Luo J, Shi X. Solutions to elliptic systems involving doubly critical nonlinearities and Hardy-type potentials. Acta Math Sci, 2015, 35B: 423-438
[27] Kang D, Yang F. Elliptic systems involving multiple critical nonlinearities and symmetric multi-polar potentials. Sci China Math, 2014, 57: 1011-1024
[28] Lieb E. Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann of Math, 1983, 118: 349-374
[29] Palatucci G, Pisante A. Improved Sobolev embeddings, profile decomposition,concentration-compactness for fractional Sobolev spaces. Calc Var Partial Differential Equations, 2014, 50: 799-829
[30] Serrin J. Local behavior of solutions of quasi-linear equations. Acta Math, 1964, 111: 247-302
[31] Struwe M. Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Berlin: Springer-Verlag, 2000
[32] Su Y, Feng Z. Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential. Discrete and Continuous Dynamical Systems-S, 2023, 16: 589-601
[33] Talenti G. Best constant in Sobolev inequality. Ann Mat Pura Appl, 1976, 110: 353-372
[34] Terracini S. On positive entire solutions to a class of equations with a singular cofficient and critical exponent. Adv Differential Equations, 1996, 1: 241-264
[35] Tolksdorf P. Regularity for a more general class of quasilinear elliptic equations. J Differential Equations, 1984, 51: 126-150
[36] Vàzquez J. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim, 1984, 12: 191-202
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