June  2019, 39(6): 3345-3364. doi: 10.3934/dcds.2019138

Hardy-Sobolev type inequality and supercritical extremal problem

1. 

Department of Mathematics, Federal University of Piauí, 64049-550 Teresina, PI, Brazil

2. 

Department of Mathematics, University of Brasília, 70910-900, Brasília, DF, Brazil

3. 

Departamento de Matematica, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile

* Corresponding author: João Marcos do Ó

Received  July 2018 Revised  December 2018 Published  February 2019

Fund Project: The third author was supported by FONDECYT grants 1181125, 1161635 and 1171691.

This paper deals with Hardy-Sobolev type inequalities involving variable exponents. Our approach also enables us to prove existence results for a wide class of quasilinear elliptic equations with supercritical power-type nonlinearity with variable exponent.

Citation: José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
References:
[1]

A. BalinskyW. D. Evans and R. T. Lewis, Hardy's inequality and curvature, J. Funct. Anal., 262 (2012), 648-666.  doi: 10.1016/j.jfa.2011.10.001.  Google Scholar

[2]

E. BerchioD. Ganguly and G. Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space, J. Funct. Anal., 272 (2017), 1661-1703.  doi: 10.1016/j.jfa.2016.11.018.  Google Scholar

[3]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[4]

H. BrezisM. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight, J. Funct. Anal., 171 (2000), 177-191.  doi: 10.1006/jfan.1999.3504.  Google Scholar

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[6]

P. ClémentD. G. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal., 7 (1996), 133-170.  doi: 10.12775/TMNA.1996.006.  Google Scholar

[7]

A. Cotsiolis and N. Labropoulos, Sharp Hardy inequalities on the solid torus, J. Math. Anal. Appl., 448 (2017), 841-863.  doi: 10.1016/j.jmaa.2016.11.042.  Google Scholar

[8]

D. G. de FigueiredoJ. V. Gonçalves and O. H. Miyagaki, On a class of quasilinear elliptic problems involving critical exponents, Commun. Contemp. Math., 2 (2000), 47-59.  doi: 10.1142/S0219199700000049.  Google Scholar

[9]

J. F. de Oliveira and J. M. do Ó, Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions, Proc. Amer. Math. Soc., 142 (2014), 2813-2828.  doi: 10.1090/S0002-9939-2014-12019-3.  Google Scholar

[10]

J. F. de Oliveira, On a class of quasilinear elliptic problems with critical exponential growth on the whole space, Topol. Methods Nonlinear Anal., 49 (2017), 529-550.  doi: 10.12775/TMNA.2016.086.  Google Scholar

[11]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[12]

J. M. do Ó and J. F. de Oliveira, Concentration-compactness and extremal problems for a weighted Trudinger-Moser inequality, Commun. Contemp. Math., 19 (2017), 1650003, 26pp. doi: 10.1142/S0219199716500036.  Google Scholar

[13]

J. M. do Ó, B. Ruf and P. Ubilla, On supercritical Sobolev type inequalities and related elliptic equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 83, 18 pp. doi: 10.1007/s00526-016-1015-6.  Google Scholar

[14]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[15]

G. H. Hardy, Note on a theorem of Hilbert, Math. Z., 6 (1920), 314-317.  doi: 10.1007/BF01199965.  Google Scholar

[16] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, University Press, ${2^{\mathit{nd}}}$ edition, Cambridge, at the University Press, 1952.   Google Scholar
[17]

J. Jacobsen and K. Schmitt, Radial solutions of quasilinear elliptic differential equations, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, (2004), 359-435.  Google Scholar

[18]

J. Jacobsen and K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations, 184 (2002), 283-298.  doi: 10.1006/jdeq.2001.4151.  Google Scholar

[19]

A. Kufner and B. Opic, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific and Technical, Harlow, 1990.  Google Scholar

[20]

E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki, 67 (2000), 563-572.  doi: 10.1007/BF02676404.  Google Scholar

[21]

D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and its Applications (East European Series), 53, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3562-7.  Google Scholar

[22]

D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis, Mathematics and its Applications (East European Series), 61, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1043-5.  Google Scholar

[23]

B. G. Pachpatte, On some generalizations of Hardy's integral inequality, J. Math. Anal. Appl., 234 (1999), 15-30.  doi: 10.1006/jmaa.1999.6294.  Google Scholar

[24]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[25]

G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Advances in Mathematics, 230 (2012), 294-320.  doi: 10.1016/j.aim.2011.12.001.  Google Scholar

show all references

References:
[1]

A. BalinskyW. D. Evans and R. T. Lewis, Hardy's inequality and curvature, J. Funct. Anal., 262 (2012), 648-666.  doi: 10.1016/j.jfa.2011.10.001.  Google Scholar

[2]

E. BerchioD. Ganguly and G. Grillo, Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space, J. Funct. Anal., 272 (2017), 1661-1703.  doi: 10.1016/j.jfa.2016.11.018.  Google Scholar

[3]

H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[4]

H. BrezisM. Marcus and I. Shafrir, Extremal functions for Hardy's inequality with weight, J. Funct. Anal., 171 (2000), 177-191.  doi: 10.1006/jfan.1999.3504.  Google Scholar

[5]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.  Google Scholar

[6]

P. ClémentD. G. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal., 7 (1996), 133-170.  doi: 10.12775/TMNA.1996.006.  Google Scholar

[7]

A. Cotsiolis and N. Labropoulos, Sharp Hardy inequalities on the solid torus, J. Math. Anal. Appl., 448 (2017), 841-863.  doi: 10.1016/j.jmaa.2016.11.042.  Google Scholar

[8]

D. G. de FigueiredoJ. V. Gonçalves and O. H. Miyagaki, On a class of quasilinear elliptic problems involving critical exponents, Commun. Contemp. Math., 2 (2000), 47-59.  doi: 10.1142/S0219199700000049.  Google Scholar

[9]

J. F. de Oliveira and J. M. do Ó, Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions, Proc. Amer. Math. Soc., 142 (2014), 2813-2828.  doi: 10.1090/S0002-9939-2014-12019-3.  Google Scholar

[10]

J. F. de Oliveira, On a class of quasilinear elliptic problems with critical exponential growth on the whole space, Topol. Methods Nonlinear Anal., 49 (2017), 529-550.  doi: 10.12775/TMNA.2016.086.  Google Scholar

[11]

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics 2017, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.  Google Scholar

[12]

J. M. do Ó and J. F. de Oliveira, Concentration-compactness and extremal problems for a weighted Trudinger-Moser inequality, Commun. Contemp. Math., 19 (2017), 1650003, 26pp. doi: 10.1142/S0219199716500036.  Google Scholar

[13]

J. M. do Ó, B. Ruf and P. Ubilla, On supercritical Sobolev type inequalities and related elliptic equations, Calc. Var. Partial Differential Equations, 55 (2016), Art. 83, 18 pp. doi: 10.1007/s00526-016-1015-6.  Google Scholar

[14]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[15]

G. H. Hardy, Note on a theorem of Hilbert, Math. Z., 6 (1920), 314-317.  doi: 10.1007/BF01199965.  Google Scholar

[16] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, University Press, ${2^{\mathit{nd}}}$ edition, Cambridge, at the University Press, 1952.   Google Scholar
[17]

J. Jacobsen and K. Schmitt, Radial solutions of quasilinear elliptic differential equations, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, (2004), 359-435.  Google Scholar

[18]

J. Jacobsen and K. Schmitt, The Liouville-Bratu-Gelfand problem for radial operators, J. Differential Equations, 184 (2002), 283-298.  doi: 10.1006/jdeq.2001.4151.  Google Scholar

[19]

A. Kufner and B. Opic, Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, vol. 219, Longman Scientific and Technical, Harlow, 1990.  Google Scholar

[20]

E. Mitidieri, A simple approach to Hardy inequalities, Mat. Zametki, 67 (2000), 563-572.  doi: 10.1007/BF02676404.  Google Scholar

[21]

D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and its Applications (East European Series), 53, Kluwer Academic Publishers Group, Dordrecht, 1991. doi: 10.1007/978-94-011-3562-7.  Google Scholar

[22]

D. S. Mitrinovic, J. E. Pecaric and A. M. Fink, Classical and New Inequalities in Analysis, Mathematics and its Applications (East European Series), 61, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-017-1043-5.  Google Scholar

[23]

B. G. Pachpatte, On some generalizations of Hardy's integral inequality, J. Math. Anal. Appl., 234 (1999), 15-30.  doi: 10.1006/jmaa.1999.6294.  Google Scholar

[24]

W. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.  doi: 10.1007/BF01626517.  Google Scholar

[25]

G. Wang and D. Ye, A Hardy-Moser-Trudinger inequality, Advances in Mathematics, 230 (2012), 294-320.  doi: 10.1016/j.aim.2011.12.001.  Google Scholar

[1]

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 & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[2]

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

[3]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[4]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[5]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[6]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020031

[7]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[8]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[9]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[10]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[11]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[12]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[13]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[14]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274

[15]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[16]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[17]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[18]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[19]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355

[20]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (124)
  • HTML views (186)
  • Cited by (3)

[Back to Top]