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

Hardy-Sobolev type inequality and supercritical extremal problem

José Francisco de Oliveira 1, , João Marcos do Ó 2,, and  Pedro Ubilla 3,

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.

Keywords: Hardy-type inequality, critical exponents, supercritical growth, extremal problem, Sobolev space.
Mathematics Subject Classification: Primary: 46E35, 26D10, 35J62; Secondary: 35B33.
Citation: José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
References:
[1]

A. Balinsky, W. 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.

[2]

E. Berchio, D. 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.

[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.

[4]

H. Brezis, M. 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.

[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.

[6]

P. Clément, D. 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.

[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.

[8]

D. G. de Figueiredo, J. 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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

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

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

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[24]

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

[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.

show all references

References:
[1]

A. Balinsky, W. 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.

[2]

E. Berchio, D. 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.

[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.

[4]

H. Brezis, M. 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.

[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.

[6]

P. Clément, D. 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.

[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.

[8]

D. G. de Figueiredo, J. 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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

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

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

[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.

[19]

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

[20]

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

[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.

[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.

[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.

[24]

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

[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.

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