Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential
Merchán, Susana ; Montoro, Luigi ; Peral, Ireneo ; Sciunzi, Berardino
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014), p. 1-22 / Harvested from Numdam
Access to full text

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 0∈Ω, 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.

Publié le : 2014-01-01
DOI : https://doi.org/10.1016/j.anihpc.2013.01.003
Classification:  35J20,  35J25,  35J62,  35J70,  35J92,  46E30,  46E35 
@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--Leray potential},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {31},
     year = {2014},
     pages = {1-22},
     doi = {10.1016/j.anihpc.2013.01.003},
     mrnumber = {3165277},
     zbl = {1291.35082},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 1-22. doi : 10.1016/j.anihpc.2013.01.003. http://gdmltest.u-ga.fr/item/AIHPC_2014__31_1_1_0/

[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 | Numdam | MR 1354907 | Zbl 0866.35037

[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 1159383 | Zbl 0784.35025

[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 1183665 | Zbl 0783.35020

[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 | Numdam | MR 764943 | Zbl 0557.35051

[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 1441620 | Zbl 0869.35050

[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 | Numdam | MR 1760541 | Zbl 0958.35045

[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 | Numdam | MR 1648566 | Zbl 0930.35070

[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 1776351 | Zbl 1002.35045

[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 2096703 | Zbl 1108.35069

[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 2796244 | Zbl 1232.35048

[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 709038 | Zbl 0539.35027

[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 544879 | Zbl 0425.35020

[13] J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, Oxford (1993) | MR 1207810 | Zbl 0776.31007

[14] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 no. 11 (1988), 1203-1219 | MR 969499 | Zbl 0675.35042

[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 2926226 | Zbl 1248.35081

[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 3180936

[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 2683684 | Zbl 1253.35009

[18] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 no. 1 (1984), 126-150 | MR 727034 | Zbl 0488.35017

[19] P. Pucci, J. Serrin, The Maximum Principle, Birkhäuser, Boston (2007) | MR 2356201 | Zbl 1134.35001

[20] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247-302 | MR 170096 | Zbl 0128.09101

[21] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 no. 4 (1971), 304-318 | MR 333220 | Zbl 0222.31007