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Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term
On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities
1. | Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto Sumiyoshi-ku, Osaka-shi, Osaka 558-8585 Japan |
2. | Department of Mathematics, Institute of Applied Mathematical Sciences, National Center for Theoretical Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd, Taipei 10617, Taiwan |
3. | National Center for Theoretical Sciences, No. 1 Sec. 4 Roosevelt Rd., National Taiwan University, Taipei, 10617, Taiwan |
$N ≥ 3$ |
$Ω \subset \mathbb{R}^N$ |
$C^2$ |
$u ∈ H^1(Ω)$ |
$\begin{align*}\left\{\begin{array}{l}-\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \tau \frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\text{ in }\Omega\\\frac{\partial u}{\partial \nu} = 0 \text{ on }\partial\Omega,\end{array}\right.\end{align*}$ |
$τ = 1$ |
$-1$ |
$0 < s <2$ |
$2^*(s) = \frac{2(N-s)}{N-2}$ |
$x_1, x_2 ∈ \overline{Ω}$ |
$x_1 ≠ x_2$ |
$λ$ |
$λ > 0$ |
$s$ |
$τ$ |
$N$ |
References:
[1] |
T. Bartsch, S. Peng and Z. Zhang,
Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Diff. Equ., 30 (2007), 113-136.
doi: 10.1007/s00526-006-0086-1. |
[2] |
G. Cerami, X. Zhong and W. Zou,
On some nonlinear elliptic PDEs with Sobolev-Hardy critical exponents and a Li-Lin open problem, Calc. Var. Partial Diff. Equ., 54 (2015), 1793-1829.
doi: 10.1007/s00526-015-0844-z. |
[3] |
J. Chabrowski,
On the Neumann problem with the Hardy-Sobolev potential, Annali di Matematica, 186 (2007), 703-719.
doi: 10.1007/s10231-006-0027-9. |
[4] |
N. Ghoussoub and X. S. Kang,
Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 767-793.
doi: 10.1016/j.anihpc.2003.07.002. |
[5] |
N. Ghoussoub and F. Robert,
Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth, IMRP Int. Math. Res. Pap., 21867 (2006), 1-85.
|
[6] |
N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities, Geom. Funct. Anal. 16 (2006), 1201-1245.
doi: 10.1007/s00039-006-0579-2. |
[7] |
N. Ghoussoub and C. Yuan,
Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.
doi: 10.1090/S0002-9947-00-02560-5. |
[8] |
M. Hashizume,
Asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy-Sobolev critical exponent, J. Differential Equations, 262 (2017), 3107-3131.
doi: 10.1016/j.jde.2016.11.005. |
[9] |
C. Hsia, C. Lin and H. Wadade,
Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal., 259 (2010), 1816-1849.
doi: 10.1016/j.jfa.2010.05.004. |
[10] |
Y. Li and C.-S. Lin,
A nonlinear elliptic pde with two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal, 203 (2012), 943-968.
doi: 10.1007/s00205-011-0467-2. |
[11] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequal ities, Ann. of Math, 118 (1983), 349-374.
doi: 10.2307/2007032. |
[12] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[13] |
R. Musina,
Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal, 68 (2008), 3972-3986.
doi: 10.1016/j.na.2007.04.034. |
[14] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-02624-3. |
[15] |
J.-L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[16] |
X. J. Wang,
Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.
doi: 10.1016/0022-0396(91)90014-Z. |
[17] |
X.-X. Zhong and W.-M. Zou, A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\mathbb{R}^N$, arXiv: 1504.01133. |
show all references
References:
[1] |
T. Bartsch, S. Peng and Z. Zhang,
Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Diff. Equ., 30 (2007), 113-136.
doi: 10.1007/s00526-006-0086-1. |
[2] |
G. Cerami, X. Zhong and W. Zou,
On some nonlinear elliptic PDEs with Sobolev-Hardy critical exponents and a Li-Lin open problem, Calc. Var. Partial Diff. Equ., 54 (2015), 1793-1829.
doi: 10.1007/s00526-015-0844-z. |
[3] |
J. Chabrowski,
On the Neumann problem with the Hardy-Sobolev potential, Annali di Matematica, 186 (2007), 703-719.
doi: 10.1007/s10231-006-0027-9. |
[4] |
N. Ghoussoub and X. S. Kang,
Hardy-Sobolev critical elliptic equations with boundary singularities, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 767-793.
doi: 10.1016/j.anihpc.2003.07.002. |
[5] |
N. Ghoussoub and F. Robert,
Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth, IMRP Int. Math. Res. Pap., 21867 (2006), 1-85.
|
[6] |
N. Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities, Geom. Funct. Anal. 16 (2006), 1201-1245.
doi: 10.1007/s00039-006-0579-2. |
[7] |
N. Ghoussoub and C. Yuan,
Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.
doi: 10.1090/S0002-9947-00-02560-5. |
[8] |
M. Hashizume,
Asymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy-Sobolev critical exponent, J. Differential Equations, 262 (2017), 3107-3131.
doi: 10.1016/j.jde.2016.11.005. |
[9] |
C. Hsia, C. Lin and H. Wadade,
Revisiting an idea of Brézis and Nirenberg, J. Funct. Anal., 259 (2010), 1816-1849.
doi: 10.1016/j.jfa.2010.05.004. |
[10] |
Y. Li and C.-S. Lin,
A nonlinear elliptic pde with two Sobolev-Hardy critical exponents, Arch. Ration. Mech. Anal, 203 (2012), 943-968.
doi: 10.1007/s00205-011-0467-2. |
[11] |
E. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequal ities, Ann. of Math, 118 (1983), 349-374.
doi: 10.2307/2007032. |
[12] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The limit case. Ⅰ, Rev. Mat. Iberoamericana, 1 (1985), 145-201.
doi: 10.4171/RMI/6. |
[13] |
R. Musina,
Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Anal, 68 (2008), 3972-3986.
doi: 10.1016/j.na.2007.04.034. |
[14] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-02624-3. |
[15] |
J.-L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[16] |
X. J. Wang,
Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991), 283-310.
doi: 10.1016/0022-0396(91)90014-Z. |
[17] |
X.-X. Zhong and W.-M. Zou, A nonlinear elliptic PDE with multiple Hardy-Sobolev critical exponents in $\mathbb{R}^N$, arXiv: 1504.01133. |
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