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 and 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

[1]

Jinbo Geng, Xiaochun Chen, Sadek Gala. On regularity criteria for the 3D magneto-micropolar fluid equations in the critical Morrey-Campanato space. Communications on Pure and Applied Analysis, 2011, 10 (2) : 583-592. doi: 10.3934/cpaa.2011.10.583

[2]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237

[3]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469

[4]

Hongmei Cao, Hao-Guang Li, Chao-Jiang Xu, Jiang Xu. Well-posedness of Cauchy problem for Landau equation in critical Besov space. Kinetic and Related Models, 2019, 12 (4) : 829-884. doi: 10.3934/krm.2019032

[5]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215

[6]

Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603

[7]

Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, 2021, 29 (3) : 2475-2488. doi: 10.3934/era.2020125

[8]

Qi-Lin Xie, Xing-Ping Wu, Chun-Lei Tang. Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2773-2786. doi: 10.3934/cpaa.2013.12.2773

[9]

Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure and Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225

[10]

Sigmund Selberg. Global existence in the critical space for the Thirring and Gross-Neveu models coupled with the electromagnetic field. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2555-2569. doi: 10.3934/dcds.2018107

[11]

Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072

[12]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[13]

P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure and Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691

[14]

Alexander Blokh, Clinton Curry, Lex Oversteegen. Density of orbits in laminations and the space of critical portraits. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2027-2039. doi: 10.3934/dcds.2012.32.2027

[15]

Marco Bravin, Luis Vega. On the one dimensional cubic NLS in a critical space. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2563-2584. doi: 10.3934/dcds.2021203

[16]

Ryan Hynd, Francis Seuffert. On the symmetry and monotonicity of Morrey extremals. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5285-5303. doi: 10.3934/cpaa.2020238

[17]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[18]

Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017

[19]

Xiaoliang Li, Cong Wang. An optimization problem in heat conduction with volume constraint and double obstacles. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022084

[20]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (274)
  • HTML views (88)
  • Cited by (0)

Other articles
by authors

[Back to Top]