Local properties of solutions of a~semilinear elliptic equation with an inverse-square potential
Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 29-43.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the behavior at the origin of distribution solutions to a semilinear elliptic equation with an inverse-square potential. The relationship with non-uniqueness of solutions to the companion parabolic equation is discussed.
@article{CMFD_2006_17_a2,
     author = {A. Tesei},
     title = {Local properties of solutions of a~semilinear elliptic equation with an inverse-square potential},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {29--43},
     publisher = {mathdoc},
     volume = {17},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2006_17_a2/}
}
TY  - JOUR
AU  - A. Tesei
TI  - Local properties of solutions of a~semilinear elliptic equation with an inverse-square potential
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2006
SP  - 29
EP  - 43
VL  - 17
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2006_17_a2/
LA  - ru
ID  - CMFD_2006_17_a2
ER  - 
%0 Journal Article
%A A. Tesei
%T Local properties of solutions of a~semilinear elliptic equation with an inverse-square potential
%J Contemporary Mathematics. Fundamental Directions
%D 2006
%P 29-43
%V 17
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2006_17_a2/
%G ru
%F CMFD_2006_17_a2
A. Tesei. Local properties of solutions of a~semilinear elliptic equation with an inverse-square potential. Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 3, Tome 17 (2006), pp. 29-43. http://geodesic.mathdoc.fr/item/CMFD_2006_17_a2/

[1] Baras P., Goldstein J., “The heat equation with a singular potential”, Trans. Amer. Math. Soc., 284 (1984), 121–139 | DOI | MR | Zbl

[2] Brezis H., Dupaigne L., Tesei A., “On a semilinear elliptic equation with inverse-square potential”, Selecta Math., 11 (2005), 1–7 | DOI | MR | Zbl

[3] Brezis H., Lions P.-L., “A note on isolated singularities for linear elliptic equations”, Math. Anal. Appl. Part A, Adv. Math. Suppl. Studies, 7, 1981, 263–266 | MR

[4] Dupaigne L., “A nonlinear elliptic PDE with the inverse square potential”, J. Anal. Math., 86 (2002), 359–398 | DOI | MR | Zbl

[5] Haraux A., Weissler F., “Nonuniqueness for a semilinear initial value problem”, Indiana Univ. Math. J., 31 (1982), 167–189 | DOI | MR | Zbl

[6] Jannelli E., “The role played by space dimension in elliptic critical problems”, J. Differential Equations, 156 (1999), 407–426 | DOI | MR | Zbl

[7] Joseph D. D., Lundgren T. S., “Quasilinear Dirichlet problems driven by positive sources”, Arch. Ration. Mech. Anal., 49 (1973), 241–269 | MR | Zbl

[8] Kato T., “Schrödinger operators with singular potentials”, Israel J. Math., 13 (1972), 135–148 | DOI | MR

[9] Kersner R., Tesei A., “Well-posedness of initial value problems for singular parabolic equations”, J. Differential Equations, 199 (2004), 47–76 | DOI | MR | Zbl

[10] Lions P.-L., “Isolated singularities in semilinear problems”, J. Differential Equations, 38 (1980), 441–450 | DOI | MR | Zbl

[11] Moschini L., Reyes G., Tesei A., “Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients”, Commun. Pure Appl. Anal. (to appear) | MR

[12] Moschini L., Tesei A., “A parabolic Harnack inequality for the heat equation with inverse-square potential”, Forum Math. (to appear) | MR

[13] Ni W.-M., Sacks P., “Singular behavior in nonlinear parabolic equations”, Trans. Amer. Math. Soc., 287 (1985), 657–671 | DOI | MR | Zbl

[14] Pohozaev S. I., Tesei A., “Nonexistence of local solutions to semilinear partial differential inequalities”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21:4 (2004), 487–502 | DOI | MR | Zbl

[15] Reyes G., Tesei A., “Self-similar solutions of a semilinear parabolic equation with inverse-square potential”, J. Differential Equations (to appear) | MR

[16] Serrin J., “Pathological solutions of elliptic differential equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17 (1964), 385–387 | MR

[17] Smets D., Tesei A., “On a class of semilinear elliptic problems with first order terms”, Adv. Differential Equations, 8 (2003), 257–278 | MR | Zbl

[18] Talenti G., “Best constant in Sobolev inequality”, Ann. Mat. Pura Appl. (4), 110 (1976), 353–372 | DOI | MR | Zbl

[19] Terracini S., “On positive solutions to a class of equations with singular coefficient and critical exponent”, Adv. Differential Equations, 1 (1996), 241–264 | MR | Zbl

[20] Vazquez J. L., Zuazua E., “The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential”, J. Funct. Anal., 173 (2000), 103–153 | DOI | MR | Zbl

[21] Weissler F. B., “Local existence and nonexistence for semilinear parabolic equations in $L^p$”, Indiana Univ. Math. J., 29 (1980), 79–102 | DOI | MR | Zbl