| Title:
|
A refinement of the radial Pohozaev identity (English) |
| Author:
|
Catrina, Florin |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
135 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
143-150 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this article we produce a refined version of the classical Pohozaev identity in the radial setting. The refined identity is then compared to the original, and possible applications are discussed. (English) |
| Keyword:
|
Green's function |
| Keyword:
|
positive solutions |
| Keyword:
|
supercritical nonlinearity |
| MSC:
|
35J25 |
| MSC:
|
35J60 |
| MSC:
|
35J70 |
| idZBL:
|
Zbl 1224.35090 |
| idMR:
|
MR2723081 |
| DOI:
|
10.21136/MB.2010.140691 |
| . |
| Date available:
|
2010-07-20T18:32:02Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140691 |
| . |
| Reference:
|
[1] Brezis, H.: Elliptic equations with limiting Sobolev exponents---the impact of topology.Comm. Pure Appl. Math. 39 (1986), S17--S39. Zbl 0601.35043, MR 0861481, 10.1002/cpa.3160390704 |
| Reference:
|
[2] Brezis, H., Dupaigne, L., Tesei, A.: On a semilinear elliptic equation with inverse-square potential.Selecta Math. 11 (2005), 1-7. Zbl 1161.35383, MR 2179651, 10.1007/s00029-005-0003-z |
| Reference:
|
[3] Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent.Comm. Pure Appl. Math. 36 (1983), 437-477. MR 0709644, 10.1002/cpa.3160360405 |
| Reference:
|
[4] Brezis, H., Peletier, L. A.: Elliptic equations with critical exponent on $S\sp 3$: new non-minimising solutions.C. R. Math. Acad. Sci. Paris 339 (2004), 391-394. MR 2092750, 10.1016/j.crma.2004.07.010 |
| Reference:
|
[5] Brezis, H., Willem, M.: On some nonlinear equations with critical exponents.J. Funct. Anal. 255 (2008), 2286-2298. Zbl 1166.35021, MR 2473258, 10.1016/j.jfa.2007.11.022 |
| Reference:
|
[6] Budd, C. J., Humphries, A. R.: Numerical and analytical estimates of existence regions for semi-linear elliptic equations with critical Sobolev exponents in cuboid and cylindrical domains.J. Comput. Appl. Math. 151 (2003), 59-84. Zbl 1016.65082, MR 1950229, 10.1016/S0377-0427(02)00737-9 |
| Reference:
|
[7] Catrina, F.: A note on a result of M. Grossi.Proc. Amer. Soc. 137 (2009), 3717-3724. Zbl 1179.35142, MR 2529879, 10.1090/S0002-9939-09-10031-X |
| Reference:
|
[8] Catrina, F., Lavine, R.: Radial solutions for weighted semilinear equations.Commun. Contemp. Math. 4 (2002), 529-545. Zbl 1013.35023, MR 1918758, 10.1142/S0219199702000750 |
| Reference:
|
[9] Chou, K.-S., Geng, D.: On the critical dimension of a semilinear degenerate elliptic equation involving critical Sobolev-Hardy exponent.Nonlinear Anal., Theory Methods Appl. 12 (1996), 1965-1984. Zbl 0855.35042, MR 1386127, 10.1016/0362-546X(95)00048-Z |
| Reference:
|
[10] Clement, Ph., Figueiredo, D. G. de, Mitidieri, E.: Quasilinear elliptic equations with critical exponents.Topol. Methods Nonlinear Anal. 7 (1996), 133-170. Zbl 0939.35072, MR 1422009, 10.12775/TMNA.1996.006 |
| Reference:
|
[11] Dávila, J., Pino, M. del, Musso, M., Wei, J.: Fast and slow decay solutions for supercritical elliptic problems in exterior domains.Calc. Var. Partial Differential Equations 32 (2008), 453-480. MR 2402919, 10.1007/s00526-007-0154-1 |
| Reference:
|
[12] Druet, O.: Elliptic equations with critical Sobolev exponents in dimension 3.Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 125-142. Zbl 1011.35060, MR 1902741, 10.1016/S0294-1449(02)00095-1 |
| Reference:
|
[13] Egnell, H.: Semilinear elliptic equations involving critical Sobolev exponents.Arch. Rational Mech. Anal. 104 (1988), 27-56. Zbl 0674.35033, MR 0956566, 10.1007/BF00256931 |
| Reference:
|
[14] Egnell, H.: Existence and nonexistence results for $m$-Laplace equations involving critical Sobolev exponents.Arch. Rational Mech. Anal. 104 (1988), 57-77. Zbl 0675.35036, MR 0956567, 10.1007/BF00256932 |
| Reference:
|
[15] Felmer, P. L., Quaas, A.: Positive radial solutions to a `semilinear' equation involving the Pucci's operator.J. Differ. Equations 199 (2004), 376-393. MR 2047915, 10.1016/j.jde.2004.01.001 |
| Reference:
|
[16] Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDE's involving the critical Sobolev and Hardy exponents.Trans. Amer. Math. Soc. 352 (2000), 5703-5743. MR 1695021, 10.1090/S0002-9947-00-02560-5 |
| Reference:
|
[17] Gidas, B., Ni, W. M., Nirenberg, L.: Symmetry and related properties via the maximum principle.Comm. Math. Phys. 68 (1979), 209-243. Zbl 0425.35020, MR 0544879, 10.1007/BF01221125 |
| Reference:
|
[18] Grossi, M.: Radial solutions for the Brezis-Nirenberg problem involving large nonlinearities.J. Funct. Anal. 254 (2008), 2995-3036. Zbl 1154.35050, MR 2418617, 10.1016/j.jfa.2008.03.007 |
| Reference:
|
[19] Grossi, M.: Existence of radial solutions for an elliptic problem involving exponential nonlinearities.Discrete Contin. Dyn. Syst. 21 (2008), 221-232. Zbl 1155.35042, MR 2379462 |
| Reference:
|
[20] Jacobsen, J.: Global bifurcation problems associated with $k$-Hessian operators.Topol. Methods Nonlinear Anal. 14 (1999), 81-130. Zbl 0959.35066, MR 1758881, 10.12775/TMNA.1999.023 |
| Reference:
|
[21] Passaseo, D.: Some sufficient conditions for the existence of positive solutions to the equation $-\Delta u+a(x)u=u\sp {2\sp *-1}$ in bounded domains.Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 185-227. MR 1378466, 10.1016/S0294-1449(16)30102-0 |
| Reference:
|
[22] Pohozaev, S. I.: On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$.Russian Dokl. Akad. Nauk SSSR 165 (1965), 36-39. MR 0192184 |
| Reference:
|
[23] Pucci, P., Serrin, J.: Critical exponents and critical dimensions for polyharmonic operators.J. Math. Pures Appl. 69 (1990), 55-83. MR 1054124 |
| Reference:
|
[24] Rey, O.: The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent.J. Funct. Anal. 89 (1990), 1-52. Zbl 0786.35059, MR 1040954, 10.1016/0022-1236(90)90002-3 |
| Reference:
|
[25] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature.J. Differential Geom. 20 (1984), 479-495. Zbl 0576.53028, MR 0788292, 10.4310/jdg/1214439291 |
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