July  2020, 19(7): 3673-3695. doi: 10.3934/cpaa.2020162

Improved Sobolev inequalities and critical problems

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

*Corresponding author

Received  August 2019 Revised  January 2020 Published  April 2020

Fund Project: X. Chen is supported by NNSF of China(No:11961032), NNSF of Jiangxi Province(No:20192 BAB201003). J. Yang is supported NNSF of China(No:11671179 and 11771300)

In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in
$ \mathbb{R}^N $
involving
$ p $
-Laplacian
$ \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*} $
where
$ 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}\} $
and
$ p^\ast_{\alpha} = \frac{p(N-\alpha)}{N-p} $
is the critical Hardy-Sobolev exponent, and critical problems involving fractional Laplacian
$ \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*} $
where
$ 0<s<\frac{N}2,0<\alpha,\beta<\min\{K,\frac{2NKs}{N^2-2s(N-K)}\} $
and
$ 2^\ast_{s,\alpha} = \frac{2(N-\alpha)}{N-2s} $
.
Citation: Xiaoli Chen, Jianfu Yang. Improved Sobolev inequalities and critical problems. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3673-3695. doi: 10.3934/cpaa.2020162
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[4]

R. FilippucciP. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas, Vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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.  Google Scholar

[11]

E. S. NoussairC. 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.  Google Scholar

[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.  Google Scholar

[13]

S. SecchiD. 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.  Google Scholar

[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.  Google Scholar

[15]

M. Willem, Minimax Theorems, Birkh$\ddot{a}$user, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. FilippucciP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

E. S. NoussairC. 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.  Google Scholar

[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.  Google Scholar

[13]

S. SecchiD. 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.  Google Scholar

[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.  Google Scholar

[15]

M. Willem, Minimax Theorems, Birkh$\ddot{a}$user, Boston-Basel-Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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