Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

Kondratiev, Vladimir; Liskevich, Vitali; Moroz, Vitaly

Kondratiev, Vladimir ; Liskevich, Vitali ; Moroz, Vitaly

Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005), p. 25-43 / Harvested from Numdam

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Publié le : 2005-01-01

MR 2114410 | Zbl 02141610

DOI : https://doi.org/10.1016/j.anihpc.2004.03.003

@article{AIHPC_2005__22_1_25_0,
     author = {Kondratiev, Vladimir and Liskevich, Vitali and Moroz, Vitaly},
     title = {Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {22},
     year = {2005},
     pages = {25-43},
     doi = {10.1016/j.anihpc.2004.03.003},
     mrnumber = {2114410},
     zbl = {02141610},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2005__22_1_25_0}
}

Kondratiev, Vladimir; Liskevich, Vitali; Moroz, Vitaly. Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains. Annales de l'I.H.P. Analyse non linéaire, Tome 22 (2005) pp. 25-43. doi : 10.1016/j.anihpc.2004.03.003. http://gdmltest.u-ga.fr/item/AIHPC_2005__22_1_25_0/

Bibliographie

[1] Agmon S., On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in: Methods of Functional Analysis and Theory of Elliptic Equations (Naples, 1982), Liguori, Naples, 1983, pp. 19-52. | MR 819005 | Zbl 0595.58044

[2] Ancona A., First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains, J. Anal. Math. 72 (1997) 45-92. | MR 1482989 | Zbl 0944.58016

[3] Bandle C., Essén M., On positive solutions of Emden equations in cone-like domains, Arch. Rational Mech. Anal. 112 (1990) 319-338. | MR 1077263 | Zbl 0727.35051

[4] Bandle C., Levine H.A., On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc. 316 (1989) 595-622. | MR 937878 | Zbl 0693.35081

[5] Berestycki H., Capuzzo-Dolcetta I., Nirenberg L., Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994) 59-78. | MR 1321809 | Zbl 0816.35030

[6] Bidaut-Véron M.-F., Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal. 107 (1989) 293-324. | MR 1004713 | Zbl 0696.35022

[7] Bidaut-Véron M.-F., Pohozaev S., Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math. 84 (2001) 1-49. | MR 1849197 | Zbl 1018.35040

[8] Birindelli I., Mitidieri E., Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 1217-1247. | MR 1664101 | Zbl 0919.35023

[9] Dancer E.N., Sweers G., On the existence of a maximal weak solution for a semilinear elliptic equation, Differential Integral Equations 2 (1989) 533-540. | MR 996759 | Zbl 0732.35027

[10] Deng K., Levine H.A., The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000) 85-126. | MR 1742850 | Zbl 0942.35025

[11] Ding W.Y., Ni W.-M., On the elliptic equation $Δ u + K u^{(n + 2) / (n - 2)} = 0$ and related topics, Duke Math. J. 52 (1985) 485-506. | MR 792184 | Zbl 0592.35048

[12] Egnell H., Positive solutions of semilinear equations in cones, Trans. Amer. Math. Soc. 330 (1992) 191-201. | MR 1034662 | Zbl 0766.35014

[13] Fukushima M., Oshima Y., Takeda H., Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994. | MR 1303354 | Zbl 0838.31001

[14] Gidas B., Spruck J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981) 525-598. | MR 615628 | Zbl 0465.35003

[15] Gilbarg D., Trudinger N.S., Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983. | MR 737190 | Zbl 1042.35002

[16] Grigor'Yan A., Hansen W., A Liouville property for Schrödinger operators, Math. Ann. 312 (1998) 659-716. | MR 1660247 | Zbl 0938.31007

[17] Kondratiev V., Liskevich V., Sobol Z., Second-order semilinear elliptic inequalities in exterior domains, J. Differential Equations 187 (2003) 429-455. | MR 1949449 | Zbl 01884230

[18] Kondratiev V., Liskevich V., Sobol Z., Us A., Estimates of heat kernels for a class of second-order elliptic operators with applications to semi-linear inequalities in exterior domains, J. London Math. Soc. (2) 69 (2004) 107-127. | MR 2025330 | Zbl 1047.35006

[19] Laptev G.G., Absence of global positive solutions of systems of semilinear elliptic inequalities in cones, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000) 107-124, (Russian); translation in, Izv. Math. 64 (2000) 1197-1215. | MR 1817251 | Zbl 1013.35041

[20] Levine H.A., The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990) 262-288. | MR 1056055 | Zbl 0706.35008

[21] Littman W., Stampaccia G., Weinberger H.F., Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963) 43-77. | Numdam | MR 161019 | Zbl 0116.30302

[22] Murata M., On construction of Martin boundaries for second order elliptic equations, Publ. Res. Inst. Math. Sci. 26 (1990) 585-627. | MR 1081506 | Zbl 0726.31009

[23] Murata M., Semismall perturbations in the Martin theory for elliptic equations, Israel J. Math. 102 (1997) 29-60. | MR 1489100 | Zbl 0891.35013

[24] Murata M., Martin boundaries of elliptic skew products, semismall perturbations, and fundamental solutions of parabolic equations, J. Funct. Anal. 194 (2002) 53-141. | MR 1929139 | Zbl 01836687

[25] Mitidieri E., Pohožaev S.I., A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001) 1-384, (in Russian). | MR 1879326 | Zbl 1074.35500

[26] Ni W.M., On the elliptic equation $Δ u + K (x) u^{(n + 2) / (n - 2)} = 0$ , its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982) 493-529. | MR 662915 | Zbl 0496.35036

[27] Pinchover Y., On the equivalence of Green functions of second order elliptic equations in $R^{n}$ , Differential Integral Equations 5 (1992) 481-493. | MR 1157482 | Zbl 0772.35015

[28] Pinchover Y., On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994) 313-341. | Numdam | MR 1277898 | Zbl 0837.35010

[29] Pinchover Y., Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann. 314 (1999) 555-590. | MR 1704549 | Zbl 0928.35010

[30] Pinsky R.G., Positive Harmonic Functions and Diffusion, Cambridge Univ. Press, 1995. | MR 1326606 | Zbl 0858.31001

[31] Serrin J., Weinberger H.F., Isolated singularities of solutions of linear elliptic equations, Amer. J. Math. 88 (1966) 258-272. | MR 201815 | Zbl 0137.07001

[32] Serrin J., Zou H., Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002) 79-142. | MR 1946918 | Zbl 1059.35040

[33] Struwe M., Variational Methods, Springer-Verlag, Berlin, 1990. | MR 1078018 | Zbl 0746.49010

[34] Toland J.F., On positive solutions of $- Δ u = F (x, u)$ , Math. Z. 182 (1983) 351-357. | MR 696532 | Zbl 0491.35048

[35] Véron L., Singularities of Solutions of Second Order Quasilinear Equations, Longman, Harlow, 1996. | MR 1424468 | Zbl 0858.35018

[36] Vogt H., Equivalence of pointwise and global ellipticity estimates, Math. Nachr. 237 (2002) 125-128. | MR 1894356 | Zbl 01749726

[37] Zhang Qi S., An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with $Ricci \geq 0$ , Math. Ann. 316 (2000) 703-731. | MR 1758450 | Zbl 0974.53031

[38] Zhang Qi S., A Liouville type theorem for some critical semilinear elliptic equations on noncompact manifolds, Indiana Univ. Math. J. 50 (2001) 1915-1936. | MR 1889088 | Zbl 1082.35067