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Fourth-order problems with Leray-Lions type operators in variable exponent spaces
On the capacity approach to non-attainability of Hardy's inequality in $\mathbb{R}^N$
1. | Dip. di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria |
2. | RISM-Riemann International School of Mathematics, via G.B. Vico 46, 21100 - Varese, Italy |
3. | Dip. di Matematica, Università degli Studi di Milano, via C. Saldini 50, 20133 - Milano, Italy |
In this note we exploit nonlinear capacity estimates in the spirit of Mitidieri-Pohozaev [
References:
[1] |
A. Alvino,
Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital. A, 14 (1977), 148-156.
|
[2] |
C. Bennett and R. Sharpley,
Interpolation of Operators, Pure and Applied Mathematics, 129, Boston Academic Press, 1988. |
[3] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[4] |
D. Cassani, B. Ruf and C. Tarsi, Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$, preprint, (2017), arXiv: 1711.03763. |
[5] |
D. Cassani, F. Sani and C. Tarsi,
Equivalent Moser type inequalities in $\mathbb{R}^2$ and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263.
doi: 10.1016/j.jfa.2014.09.022. |
[6] |
A. Cianchi and A. Ferone,
Hardy inequalities with non-standard remainder terms, Ann. Inst. H. Poincare. An. Non Lineaire, 25 (2008), 889-906.
doi: 10.1016/j.anihpc.2007.05.003. |
[7] |
S. Costea,
Sobolev-Lorentz spaces in the Euclidean setting and counterexamples, Nonlinear Anal., 152 (2017), 149-182.
doi: 10.1016/j.na.2017.01.001. |
[8] |
E. B. Davies,
A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67.
|
[9] |
B. Devyver, M. Fraas and Y. Pinchover,
Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon, J. Funct. Anal., 266 (2014), 4422-4489.
doi: 10.1016/j.jfa.2014.01.017. |
[10] |
S. Filippas and A. Tertikas,
Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.
doi: 10.1006/jfan.2001.3900. |
[11] |
N. Ghoussoub and A. Moradifam,
Functional Inequalities: New Perspectives and New Applications. Mathematical Surveys and Monographs,
187 Amer. Math. Soc., Providence, RI, 2013.
doi: 10.1090/surv/187. |
[12] |
A. Kufner, L. Maligranda and L.-E. Persson,
The prehistory of the Hardy inequality, Amer. Math. Monthly, 113 (2006), 715-732.
doi: 10.1080/00029890.2006.11920356. |
[13] |
G. G. Lorentz,
Some new functional spaces, Ann. of Math., 51 (1950), 37-55.
doi: 10.2307/1969496. |
[14] |
V. Maz'ya,
Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second ed., Grundlehren der Mathematischen Wissenschaften 342, Springer, 2011.
doi: 10.1007/978-3-642-15564-2. |
[15] |
E. Mitidieri and S. Pohozaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
|
[16] |
B. Opic and A. Kufner,
Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219, 1990. |
show all references
References:
[1] |
A. Alvino,
Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital. A, 14 (1977), 148-156.
|
[2] |
C. Bennett and R. Sharpley,
Interpolation of Operators, Pure and Applied Mathematics, 129, Boston Academic Press, 1988. |
[3] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[4] |
D. Cassani, B. Ruf and C. Tarsi, Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$, preprint, (2017), arXiv: 1711.03763. |
[5] |
D. Cassani, F. Sani and C. Tarsi,
Equivalent Moser type inequalities in $\mathbb{R}^2$ and the zero mass case, J. Funct. Anal., 267 (2014), 4236-4263.
doi: 10.1016/j.jfa.2014.09.022. |
[6] |
A. Cianchi and A. Ferone,
Hardy inequalities with non-standard remainder terms, Ann. Inst. H. Poincare. An. Non Lineaire, 25 (2008), 889-906.
doi: 10.1016/j.anihpc.2007.05.003. |
[7] |
S. Costea,
Sobolev-Lorentz spaces in the Euclidean setting and counterexamples, Nonlinear Anal., 152 (2017), 149-182.
doi: 10.1016/j.na.2017.01.001. |
[8] |
E. B. Davies,
A review of Hardy inequalities, Oper. Theory Adv. Appl., 110 (1999), 55-67.
|
[9] |
B. Devyver, M. Fraas and Y. Pinchover,
Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon, J. Funct. Anal., 266 (2014), 4422-4489.
doi: 10.1016/j.jfa.2014.01.017. |
[10] |
S. Filippas and A. Tertikas,
Optimizing improved Hardy inequalities, J. Funct. Anal., 192 (2002), 186-233.
doi: 10.1006/jfan.2001.3900. |
[11] |
N. Ghoussoub and A. Moradifam,
Functional Inequalities: New Perspectives and New Applications. Mathematical Surveys and Monographs,
187 Amer. Math. Soc., Providence, RI, 2013.
doi: 10.1090/surv/187. |
[12] |
A. Kufner, L. Maligranda and L.-E. Persson,
The prehistory of the Hardy inequality, Amer. Math. Monthly, 113 (2006), 715-732.
doi: 10.1080/00029890.2006.11920356. |
[13] |
G. G. Lorentz,
Some new functional spaces, Ann. of Math., 51 (1950), 37-55.
doi: 10.2307/1969496. |
[14] |
V. Maz'ya,
Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Second ed., Grundlehren der Mathematischen Wissenschaften 342, Springer, 2011.
doi: 10.1007/978-3-642-15564-2. |
[15] |
E. Mitidieri and S. Pohozaev,
A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
|
[16] |
B. Opic and A. Kufner,
Hardy-type Inequalities, Pitman Research Notes in Mathematics Series, 219, 1990. |
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