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A fractional Korn-type inequality
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 |
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.
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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