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Bound state positive solutions for a class of elliptic system with Hartree nonlinearity
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Improved Sobolev inequalities and critical problems
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
$ \mathbb{R}^N $ |
$ p $ |
$ \begin{equation*} \label{eq:1} -\Delta_pu = \frac{|u|^{p^\ast_{\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{p^\ast_{\beta}-2}u}{|y|^{\beta}}, \end{equation*} $ |
$ x = (y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K},1\leq K\leq N,1<p<N,0<\alpha,\beta<\min\{K,\frac{NKp}{N^2-(N-K)p}\} $ |
$ p^\ast_{\alpha} = \frac{p(N-\alpha)}{N-p} $ |
$ \begin{equation*} (-\Delta)^{s}u = \frac{|u|^{2^\ast_{s,\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{2^\ast_{s,\beta}-2}u}{|y|^{\beta}}, \end{equation*} $ |
$ 0<s<\frac{N}2,0<\alpha,\beta<\min\{K,\frac{2NKs}{N^2-2s(N-K)}\} $ |
$ 2^\ast_{s,\alpha} = \frac{2(N-\alpha)}{N-2s} $ |
References:
[1] |
M. Badiale and G. Tarantello,
A Sobolev-Hardy inequality with applications to nonlinear elliptic equation arising in astrophisics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.
doi: 10.1007/s002050200201. |
[2] |
M. Bhakta,
A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities, Adv. Nonlinear Stud., 15 (2016), 835-848.
doi: 10.1515/ans-2015-0405. |
[3] |
X. Chen and J. Yang,
Weighted fractional Sobolev-Hardy inequality in $\mathbb{R}^N$, Adv. Nonlinear Stud., 16 (2016), 623-641.
doi: 10.1515/ans-2015-5002. |
[4] |
R. Filippucci, P. Pucci and F. Robert,
On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.
doi: 10.1016/j.matpur.2008.09.008. |
[5] |
N. Ghoussoub and F. Robert,
The effect of curvature on the best constant in the Hardy-Sobolev inequality, Geom. Funct. Anal., 16 (2006), 897-908.
doi: 10.1007/s00039-006-0579-2. |
[6] |
N. Ghoussoub and F. Robert,
Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth, Inter. Math. Res. Paper, 2006 (2006), 1-85.
doi: 10.1155/IMRP/2006/21867. |
[7] |
N. Ghoussoub and S. Shakerian,
Borderline variational problems involving fractional Laplacians and critical Singularities, Adv. Nonlinear Stud., 15 (2015), 527-555.
doi: 10.1515/ans-2015-0302. |
[8] |
N. Ghoussoub and C. Yuan,
Multiple sulutions 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. |
[9] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas, Vol. 14, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[10] |
P. L. Lions, concentration-compactness principle in the calculus of variations. The limit case, part 1 and part 2, Rev. Mat. Iberoam., 1 (1985), 145–201 and 45–121.
doi: 10.4171/RMI/6. |
[11] |
E. S. Noussair, C. A. Swanson and J. Yang,
Quasilinear elliptic problems with critical exponents, Nonlinear Anal. Theory Meth. Appl., 20 (1993), 285-301.
doi: 10.1016/0362-546X(93)90164-N. |
[12] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[13] |
S. Secchi, D. Smets and M. Willem,
Remarks on a Hardy-Sobolev inequality, C. R. Acad. Sci. Paris. Ser. I, 336 (2003), 811-815.
doi: 10.1016/S1631-073X(03)00202-4. |
[14] |
E. Swayer and R. L. Wheeden,
Weighted inequalities for fractional integrals on Euclidean and Homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.
doi: 10.2307/2374799. |
[15] |
M. Willem, Minimax Theorems, Birkh$\ddot{a}$user, Boston-Basel-Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[16] |
J. Yang,
Fractional Sobolev-Hardy inequality in $\mathbb{R}^N$, Nonlinear Anal. Theory Meth. Appl., 119 (2015), 179-185.
doi: 10.1016/j.na.2014.09.009. |
[17] |
J. Yang and F. Wu,
Doubly Critical Problems Involving Fractional Laplacians in $\mathbb{R}^N$, Adv. Nonlinear Stud., 17 (2017), 1-14.
doi: 10.1515/ans-2016-6012. |
[18] |
J. Yang and X. Zhu,
On the existence of nontrivial solutions of a quasilinear elliptic boundary value problem for unbounded domains, Acta Math. Sci., 7 (1987), 341-359.
doi: 10.1016/S0252-9602(18)30457-0. |
show all references
References:
[1] |
M. Badiale and G. Tarantello,
A Sobolev-Hardy inequality with applications to nonlinear elliptic equation arising in astrophisics, Arch. Ration. Mech. Anal., 163 (2002), 259-293.
doi: 10.1007/s002050200201. |
[2] |
M. Bhakta,
A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities, Adv. Nonlinear Stud., 15 (2016), 835-848.
doi: 10.1515/ans-2015-0405. |
[3] |
X. Chen and J. Yang,
Weighted fractional Sobolev-Hardy inequality in $\mathbb{R}^N$, Adv. Nonlinear Stud., 16 (2016), 623-641.
doi: 10.1515/ans-2015-5002. |
[4] |
R. Filippucci, P. Pucci and F. Robert,
On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl., 91 (2009), 156-177.
doi: 10.1016/j.matpur.2008.09.008. |
[5] |
N. Ghoussoub and F. Robert,
The effect of curvature on the best constant in the Hardy-Sobolev inequality, Geom. Funct. Anal., 16 (2006), 897-908.
doi: 10.1007/s00039-006-0579-2. |
[6] |
N. Ghoussoub and F. Robert,
Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth, Inter. Math. Res. Paper, 2006 (2006), 1-85.
doi: 10.1155/IMRP/2006/21867. |
[7] |
N. Ghoussoub and S. Shakerian,
Borderline variational problems involving fractional Laplacians and critical Singularities, Adv. Nonlinear Stud., 15 (2015), 527-555.
doi: 10.1515/ans-2015-0302. |
[8] |
N. Ghoussoub and C. Yuan,
Multiple sulutions 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. |
[9] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas, Vol. 14, American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/gsm/014. |
[10] |
P. L. Lions, concentration-compactness principle in the calculus of variations. The limit case, part 1 and part 2, Rev. Mat. Iberoam., 1 (1985), 145–201 and 45–121.
doi: 10.4171/RMI/6. |
[11] |
E. S. Noussair, C. A. Swanson and J. Yang,
Quasilinear elliptic problems with critical exponents, Nonlinear Anal. Theory Meth. Appl., 20 (1993), 285-301.
doi: 10.1016/0362-546X(93)90164-N. |
[12] |
G. Palatucci and A. Pisante,
Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differ. Equ., 50 (2014), 799-829.
doi: 10.1007/s00526-013-0656-y. |
[13] |
S. Secchi, D. Smets and M. Willem,
Remarks on a Hardy-Sobolev inequality, C. R. Acad. Sci. Paris. Ser. I, 336 (2003), 811-815.
doi: 10.1016/S1631-073X(03)00202-4. |
[14] |
E. Swayer and R. L. Wheeden,
Weighted inequalities for fractional integrals on Euclidean and Homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.
doi: 10.2307/2374799. |
[15] |
M. Willem, Minimax Theorems, Birkh$\ddot{a}$user, Boston-Basel-Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[16] |
J. Yang,
Fractional Sobolev-Hardy inequality in $\mathbb{R}^N$, Nonlinear Anal. Theory Meth. Appl., 119 (2015), 179-185.
doi: 10.1016/j.na.2014.09.009. |
[17] |
J. Yang and F. Wu,
Doubly Critical Problems Involving Fractional Laplacians in $\mathbb{R}^N$, Adv. Nonlinear Stud., 17 (2017), 1-14.
doi: 10.1515/ans-2016-6012. |
[18] |
J. Yang and X. Zhu,
On the existence of nontrivial solutions of a quasilinear elliptic boundary value problem for unbounded domains, Acta Math. Sci., 7 (1987), 341-359.
doi: 10.1016/S0252-9602(18)30457-0. |
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