Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 1-22.

In this work we deal with the existence and qualitative properties of the solutions to a supercritical problem involving the -Δ p (·) operator and the Hardy–Leray potential. Assuming 0Ω, we study the regularizing effect due to the addition of a first order nonlinear term, which provides the existence of solutions with a breaking of resonance. Once we have proved the existence of a solution, we study the qualitative properties of the solutions such as regularity, monotonicity and symmetry.

DOI : 10.1016/j.anihpc.2013.01.003
Classification : 35J20, 35J25, 35J62, 35J70, 35J92, 46E30, 46E35
Mots clés : Quasilinear elliptic equations, Hardy potential, Supercritical problems, Existence and nonexistence, Regularity, Symmetry of solutions
@article{AIHPC_2014__31_1_1_0,
     author = {Merch\'an, Susana and Montoro, Luigi and Peral, Ireneo and Sciunzi, Berardino},
     title = {Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the {Hardy{\textendash}Leray} potential},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1--22},
     publisher = {Elsevier},
     volume = {31},
     number = {1},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.01.003},
     mrnumber = {3165277},
     zbl = {1291.35082},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2013.01.003/}
}
TY  - JOUR
AU  - Merchán, Susana
AU  - Montoro, Luigi
AU  - Peral, Ireneo
AU  - Sciunzi, Berardino
TI  - Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2014
SP  - 1
EP  - 22
VL  - 31
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2013.01.003/
DO  - 10.1016/j.anihpc.2013.01.003
LA  - en
ID  - AIHPC_2014__31_1_1_0
ER  - 
%0 Journal Article
%A Merchán, Susana
%A Montoro, Luigi
%A Peral, Ireneo
%A Sciunzi, Berardino
%T Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential
%J Annales de l'I.H.P. Analyse non linéaire
%D 2014
%P 1-22
%V 31
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2013.01.003/
%R 10.1016/j.anihpc.2013.01.003
%G en
%F AIHPC_2014__31_1_1_0
Merchán, Susana; Montoro, Luigi; Peral, Ireneo; Sciunzi, Berardino. Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 1, pp. 1-22. doi : 10.1016/j.anihpc.2013.01.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.01.003/

[1] P. Bènilan, L. Boccardo, T. Gallout, R. Gariepy, M. Pierre, J. Vázquez, An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 22 no. 2 (1995), 241-273 | EuDML | Numdam | MR | Zbl

[2] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bull. Soc. Brasil. Mat. (N.S. 22 no. 1 (1991), 1-37 | MR | Zbl

[3] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 no. 19 (1992), 581-597 | MR | Zbl

[4] L. Boccardo, F. Murat, J.P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa 11 no. 2 (1984), 213-235 | EuDML | Numdam | MR | Zbl

[5] A. Dall'Aglio, Approximated solutions of equations with L 1 data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl. 170 no. 4 (1996), 207-240 | MR | Zbl

[6] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 28 no. 4 (1999), 741-808 | EuDML | Numdam | MR | Zbl

[7] L. Damascelli, F. Pacella, Monotonicity and symmetry of solutions of p-Laplace equations, 1<p<2, via the moving plane method, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 26 no. 4 (1998), 689-707 | EuDML | Numdam | MR | Zbl

[8] L. Damascelli, F. Pacella, Monotonicity and symmetry results for p-Laplace equations and applications, Adv. Differential Equations 5 no. 7–9 (2000), 1179-1200 | MR | Zbl

[9] L. Damascelli, B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differential Equations 206 no. 2 (2004), 483-515 | MR | Zbl

[10] J. Dávila, I. Peral, Nonlinear elliptic problems with a singular weight on the boundary, Calc. Var. Partial Differential Equations 41 no. 3–4 (2011), 567-586 | MR | Zbl

[11] E. Di Benedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 no. 8 (1983), 827-850 | MR | Zbl

[12] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 no. 3 (1979), 209-243 | MR | Zbl

[13] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford (1993) | MR | Zbl

[14] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 no. 11 (1988), 1203-1219 | MR | Zbl

[15] S. Merchán, I. Peral, Remarks on the solvability of an elliptic equation with a supercritical term involving the Hardy–Leray potential, J. Math. Anal. Appl. 394 (2012), 347-359 | MR | Zbl

[16] S. Merchán, L. Montoro, Remarks on the existence of solutions to some quasilinear elliptic problems involving the Hardy–Leray potential, Ann. Mat. Pura Appl., http://dx.doi.org/10.1007/s10231-012-0293-7. | MR

[17] L. Montoro, B. Sciunzi, M. Squassina, Asymptotic symmetry for a class of quasi-linear parabolic problems, Adv. Nonlinear Stud. 10 no. 4 (2010), 789-818 | MR | Zbl

[18] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 no. 1 (1984), 126-150 | MR | Zbl

[19] P. Pucci, J. Serrin, The Maximum Principle, Birkhäuser, Boston (2007) | MR | Zbl

[20] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302 | MR | Zbl

[21] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 no. 4 (1971), 304-318 | MR | Zbl

Cité par Sources :