|
A note on a result of M. Grossi
Author(s):
Florin
Catrina
Journal:
Proc. Amer. Math. Soc.
137
(2009),
3717-3724.
MSC (2000):
Primary 35J25, 35J70
Posted:
June 12, 2009
MathSciNet review:
2529879
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
The purpose of this note is to present a fact complementary to a result in a recent paper of M. Grossi. Making use of an energy balance identity, it is shown that the sufficient conditions for existence of solutions proved in Grossi's paper are also almost necessary.
References:
-
- 1.
- H. Brezis, Elliptic equations with limiting Sobolev exponents--the impact of topology, Comm. Pure Appl. Math., 39 (1986) S17-S39. MR 861481 (87k:58272)
- 2.
- H. Brezis, L. Dupaigne and A. Tesei, On a semilinear elliptic equation with inverse-square potential, Selecta Math., 11 (2005) 1-7. MR 2179651 (2006g:35052)
- 3.
- H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math., 36 (1983) 437-477. MR 709644 (84h:35059)
- 4.
- H. Brezis and L. A. Peletier, Elliptic equations with critical exponent on
: New non-minimising solutions, C. R. Math. Acad. Sci. Paris, 339 (2004) 391-394. MR 2092750 (2005f:35090) - 5.
- F. Catrina and R. Lavine, Radial solutions for weighted semilinear equations, Commun. Contemp. Math., 4 (2002) 529-545. MR 1918758 (2003e:35080)
- 6.
- K.-S. Chou and D. Geng, On the critical dimension of a semilinear degenerate elliptic equation involving critical Sobolev-Hardy exponent, Nonlinear Anal. TMA, 12 (1996) 1965-1984. MR 1386127 (97b:35079)
- 7.
- Ph. Clement, D. G. de Figueiredo and E. Mitidieri, Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal., 7 (1996) 133-170. MR 1422009 (97k:35072)
- 8.
- J. Dávila, M. del Pino, M. Musso and J. Wei, Fast and slow decay solutions for supercritical elliptic problems in exterior domains, Calc. Var. Partial Differential Equations, 32 (2008), no. 4, 453-480. MR 2402919 (2009b:35140)
- 9.
- O. Druet, Elliptic equations with critical Sobolev exponents in dimension
. Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), no. 2, 125-142. MR 1902741 (2003f:35104) - 10.
- H. Egnell, Semilinear elliptic equations involving critical Sobolev exponents, Arch. Rational Mech. Anal., 104 (1988) 27-56. MR 956566 (90e:35068)
- 11.
- H. Egnell, Existence and nonexistence results for
-Laplace equations involving critical Sobolev exponents, Arch. Rational Mech. Anal., 104 (1988) 57-77. MR 956567 (90e:35069) - 12.
- P. L. Felmer and A. Quaas, Positive radial solutions to a `semilinear' equation involving the Pucci's operator, J. Diff. Eqns., 199 (2004) 376-393. MR 2047915 (2004m:34040)
- 13.
- N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000) 5703-5743. MR 1695021 (2001b:35109)
- 14.
- B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979) 209-243. MR 544879 (80h:35043)
- 15.
- M. Grossi, Radial solutions for the Brezis-Nirenberg problem involving large nonlinearities, J. Funct. Anal., 254 (2008), no. 12, 2995-3036. MR 2418617 (2009c:35126)
- 16.
- J. Jacobsen, Global bifurcation problems associated with k-Hessian operators, Topol. Methods Nonlinear Anal., 14 (1999) 81-130. MR 1758881 (2001f:35126)
- 17.
- D. Passaseo, Some sufficient conditions for the existence of positive solutions to the equation
in bounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), no. 2, 185-227. MR 1378466 (97i:35055) - 18.
- S. I. Pohozaev, On the eigenfunctions of the equation
, (Russian) Dokl. Akad. Nauk SSSR, 165 (1965) 36-39. MR 0192184 (33:411) - 19.
- P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990) 55-83. MR 1054124 (91i:35065)
- 20.
- O. Rey, The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89 (1) (1990) 1-52. MR 1040954 (91b:35012)
- 21.
- R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984) 479-495. MR 788292 (86i:58137)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
35J25, 35J70
Retrieve articles in all Journals with
MSC (2000):
35J25, 35J70
Additional Information:
Florin
Catrina
Affiliation:
Department of Mathematics and Computer Science, St. John's University, Queens, New York 11439
Email:
catrinaf@stjohns.edu
DOI:
10.1090/S0002-9939-09-10031-X
PII:
S 0002-9939(09)10031-X
Keywords:
Green's function,
positive solutions,
supercritical exponent
Received by editor(s):
December 22, 2008
Posted:
June 12, 2009
Additional Notes:
The author is grateful to the anonymous referee for useful comments and suggestions
Communicated by:
Matthew J. Gursky
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|