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Existence and Liouville-type theorems for some indefinite quasilinear elliptic problems with potentials vanishing at infinity. (English) Zbl 1181.35101
Summary: We study the existence versus absence of nontrivial weak solutions for a class of indefinite quasilinear elliptic problems on unbounded domains with noncompact boundary in the presence of competing lower order nonlinearities with potentials decaying to zero at infinity.

MSC:
35J62 Quasilinear elliptic equations
35D30 Weak solutions to PDEs
35B53 Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs
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[1] Adimurthi, Hardy – sobolev inequality in \(H^1(\Omega)\) and its applications, Commun. contemp. math., 4, 409-434, (2002) · Zbl 1005.35072
[2] Ambrosetti, A.; Felli, V.; Malchiodi, A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. eur. math. soc., 7, 117-144, (2005) · Zbl 1064.35175
[3] Ambrosetti, A.; Malchiodi, A.; Ruiz, D., Recent trends on nonlinear elliptic equations on \(\mathbb{R}^N\), Rend. accad. naz. sci. XL mem. mat. appl. (5), 29, 3-13, (2005)
[4] Ambrosetti, A.; Malchiodi, A.; Ruiz, D., Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. anal. math., 98, 317-348, (2006) · Zbl 1142.35082
[5] Aronson, D.G.; Weinberger, H.F., Multidimensional nonlinear diffusion arising in population genetics, Adv. math., 30, 33-76, (1978) · Zbl 0407.92014
[6] Bahri, A.; Lions, P.L., Solutions of superlinear elliptic equations and their Morse indices, Comm. pure appl. math., 45, 1205-1215, (1992) · Zbl 0801.35026
[7] Bandle, C.; Essén, M., On positive solutions of Emden equations in cone-like domains, Arch. ration. mech. anal., 112, 319-338, (1990) · Zbl 0727.35051
[8] Berestycki, H.; Capuzzo-Dolcetta, I.; Nirenberg, L., Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. methods nonlinear anal., 4, 59-78, (1994) · Zbl 0816.35030
[9] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations, Arch. ration. mech. anal., 82, 313-375, (1983)
[10] Bianchi, G., Non-existence of positive solutions to semilinear elliptic equations on \(\mathbb{R}^N\) or \(\mathbb{R}_+^N\) through the method of moving planes, Comm. partial differential equations, 22, 1671-1690, (1997) · Zbl 0910.35048
[11] Bidaut-Véron, M.F., Local and global behavior of solutions of quasilinear equations of emden – fowler type, Arch. ration. mech. anal., 107, 293-324, (1989) · Zbl 0696.35022
[12] Bidaut-Véron, M.F., Necessary conditions of existence for an elliptic equation with source term and measure data involving p-Laplacian, Proc. 2001-luminy conf. on quasilinear elliptic and parabolic equations and systems, Electron. J. differ. equ. conf., 8, 23-34, (2002) · Zbl 1114.35316
[13] Bidaut-Véron, M.F.; Pohozaev, S.I., Nonexistence results and estimates for some nonlinear elliptic problems, J. anal. math., 84, 1-49, (2001) · Zbl 1018.35040
[14] Birindelli, I.; Demengel, F., Some Liouville theorems for the p-Laplacian, Proc. 2001-luminy conf. on quasilinear elliptic and parabolic equations and systems, Electron. J. differ. equ. conf., 8, 35-46, (2002) · Zbl 1034.35031
[15] Chandrasekhar, S., Introduction to the theory of stellar structure, (1957), Dover New York, reprinted by · Zbl 0079.23901
[16] Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke math. J., 63, 615-622, (1991) · Zbl 0768.35025
[17] Chipot, M.; Chlebik, M.; Fila, M.; Safrir, I., Existence of positive solutions of a semilinear elliptic equation in \(\mathbb{R}_+^N\) with a nonlinear boundary condition, J. math. anal. appl., 223, 429-471, (1998) · Zbl 0932.35086
[18] Ciarlet, P.G., Mathematical elasticity, vol. I. three-dimensional elasticity, (1988), North-Holland Amsterdam · Zbl 0648.73014
[19] Damascelli, L.; Farina, A.; Sciunzi, B.; Valdinoci, E., Liouville results for m-Laplace equations of lane – emden – fowler type, Ann. inst. H. Poincaré anal. non linéaire, 26, 1099-1119, (2009) · Zbl 1172.35405
[20] Damascelli, L.; Gladiali, F., Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. mat. iberoamericana, 20, 67-86, (2004) · Zbl 1330.35146
[21] Dancer, E.N.; Du, Y., Some remarks on Liouville type results for quasilinear elliptic equations, Proc. amer. math. soc., 131, 1891-1899, (2002) · Zbl 1076.35038
[22] Diaz, J.I., Nonlinear partial differential equations and free boundaries, vol. I. elliptic equations, Res. notes math., vol. 106, (1985), Pitman Boston, MA · Zbl 0595.35100
[23] Du, Y.; Guo, Z., Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. anal. math., 89, 277-302, (2003) · Zbl 1162.35028
[24] Du, Y.; Li, S., Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. differential equations, 10, 841-860, (2005) · Zbl 1161.35388
[25] Egnell, H., Positive solutions of semilinear equations in cones, Trans. amer. math. soc., 330, 191-201, (1992) · Zbl 0766.35014
[26] Esteban, M.J.; Lions, P.L., Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. roy. soc. Edinburgh sect. A, 93, 1-14, (1982) · Zbl 0506.35035
[27] Farina, A., On the classification of solutions of the lane – emden equation on unbounded domains in \(\mathbb{R}^N\), J. math. pures appl., 87, 537-561, (2007) · Zbl 1143.35041
[28] Farina, A., Liouville-type theorems for elliptic problems, (), 60-116 · Zbl 1191.35128
[29] Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. pure appl. math., 34, 525-598, (1981) · Zbl 0465.35003
[30] Holopainen, I.; Pankka, P., p-Laplace operator, quasiregular mappings and Picard-type theorems, (), 117-150 · Zbl 1161.30321
[31] Hu, B., Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential integral equations, 7, 301-313, (1994) · Zbl 0820.35062
[32] Kandilakis, D.A.; Lyberopoulos, A.N., Indefinite quasilinear elliptic problems with subcritical and supercritical nonlinearities on unbounded domains, J. differential equations, 230, 337-361, (2006) · Zbl 1134.35037
[33] Kazdan, J.L., Prescribing the curvature of a Riemannian manifold, CBMS reg. conf. ser. math., vol. 57, (1985), Amer. Math. Soc. Providence, RI · Zbl 0561.53048
[34] Kondratiev, V.; Liskevich, V.; Moroz, V., Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains, Ann. inst. H. Poincaré anal. non linéaire, 22, 25-43, (2005) · Zbl 1130.35053
[35] Kondratiev, V.; Liskevich, V.; Moroz, V.; Sobol, Z., A critical phenomenon for sublinear elliptic equations in cone-like domains, Bull. London math. soc., 37, 585-591, (2005) · Zbl 1122.35032
[36] Kondratiev, V.; Liskevich, V.; Sobol, Z., Second-order semilinear elliptic inequalities in exterior domains, J. differential equations, 187, 429-455, (2003) · Zbl 1247.35041
[37] Kondratiev, V.; Liskevich, V.; Sobol, Z., Positive solutions to semilinear and quasilinear elliptic equations on unbounded domains, (), 177-267 · Zbl 1193.35065
[38] Li, Y.; Ni, W.M., On the existence and symmetry properties of finite total mass solutions of the matukuma equation, the Eddington equation, and their generalizations, Arch. ration. mech. anal., 108, 175-194, (1989) · Zbl 0705.35039
[39] Li, Y.Y.; Zhang, L., Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. anal. math., 90, 27-87, (2003) · Zbl 1173.35477
[40] Li, Y.Y.; Zhu, M., Sharp Sobolev inequalities involving boundary terms, Geom. funct. anal., 8, 59-87, (1998) · Zbl 0901.58066
[41] Lin, C.S., On Liouville theorem and a priori estimates for the scalar curvature equations, Ann. sc. norm. super. Pisa cl. sci., 27, 107-130, (1998) · Zbl 0974.53032
[42] Lin, C.S., Liouville-type theorems for semilinear elliptic equations involving the Sobolev exponent, Math. Z., 228, 723-744, (1998) · Zbl 0915.35036
[43] Liskevich, V.; Lyakhova, S.; Moroz, V., Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains, Adv. differential equations, 11, 361-398, (2006) · Zbl 1194.35170
[44] Liskevich, V.; Lyakhova, S.; Moroz, V., Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains, J. differential equations, 232, 212-252, (2007) · Zbl 1387.35244
[45] Liskevich, V.; Skrypnik, I.I.; Skrypnik, I.V., Positive supersolutions to general nonlinear elliptic equations in exterior domains, Manuscripta math., 115, 521-538, (2004) · Zbl 1330.35153
[46] Lützen, J., Joseph Liouville 1809-1882: master of pure and applied mathematics, (1990), Springer-Verlag Berlin · Zbl 0701.01015
[47] Mitidieri, E.; Pohozaev, S.I., The absence of global positive solutions to quasilinear elliptic inequalities, Russian acad. sci. dokl. math., 57, 250-253, (1998)
[48] Mitidieri, E.; Pohozaev, S.I., Nonexistence of positive solutions for quasilinear elliptic problems on \(\mathbb{R}^N\), Proc. Steklov inst. math., 227, 1-32, (1999)
[49] Mitidieri, E.; Pohozaev, S.I., A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov inst. math., 234, 1-362, (2001) · Zbl 1074.35500
[50] Moschini, L., New Liouville theorems for linear second order degenerate elliptic equations in divergence form, Ann. inst. H. Poincaré anal. non linéaire, 22, 11-23, (2005) · Zbl 1130.35070
[51] Pélissier, M.-C.; Reynaud, L., Étude d’un modèle mathématique d’écoulement de glacier, C. R. acad. sci. Paris Sér. A, 279, 531-534, (1974) · Zbl 0327.73085
[52] Pflüger, K., Compact traces in weighted Sobolev spaces, Analysis, 18, 65-83, (1998) · Zbl 0923.46038
[53] Pflüger, K., Existence and multiplicity of solutions to a p-Laplacian equation with nonlinear boundary condition, Electron. J. differential equations, 10, 1-13, (1998) · Zbl 0892.35063
[54] Phuc, N.C.; Verbitsky, I.E., Local integral estimates and removable singularities for quasilinear and Hessian equations with nonlinear source terms, Comm. partial differential equations, 31, 1779-1791, (2006) · Zbl 1215.35071
[55] Phuc, N.C.; Verbitsky, I.E., Quasilinear and Hessian equations of lane – emden type, Ann. of math., 168, 859-914, (2008) · Zbl 1175.31010
[56] Pohozaev, S.I., On the fibering method for the solution of nonlinear boundary value problems, Proc. Steklov inst. math., 192, 157-173, (1992)
[57] Pohozaev, S.I., Nonlinear variational problems via the fibering method, (), 49-209 · Zbl 1184.35001
[58] Poláčik, P.; Quittner, P.; Souplet, P., Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: elliptic equations and systems, Duke math. J., 139, 555-579, (2007) · Zbl 1146.35038
[59] Pucci, P.; Serrin, J., The maximum principle, (2007), Birkhäuser · Zbl 1134.35001
[60] Rigoli, M.; Setti, A.G., A Liouville theorem for a class of superlinear elliptic equations on cones, Nodea nonlinear differential equations appl., 9, 15-36, (2002) · Zbl 1007.35025
[61] Schneider, M., Existence and nonexistence of positive solutions of indefinite elliptic problems in \(\mathbb{R}^N\), Adv. nonlinear stud., 3, 231-259, (2003) · Zbl 1050.35023
[62] Serrin, J., Entire solutions of quasilinear elliptic equations, J. math. anal. appl., 352, 3-14, (2009) · Zbl 1180.35243
[63] Serrin, J.; Zou, H., Cauchy – liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta math., 189, 79-142, (2002) · Zbl 1059.35040
[64] Sirakov, B., Existence and multiplicity of solutions of semilinear elliptic equations in \(\mathbb{R}^N\), Calc. var. partial differential equations, 11, 119-142, (2000) · Zbl 0977.35049
[65] Strauss, W., Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028
[66] Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. differential equations, 51, 126-150, (1984) · Zbl 0488.35017
[67] Wang, Z.-Q.; Zhu, M., Hardy inequalities with boundary terms, Electron. J. differential equations, 43, 1-8, (2003)
[68] Zhu, M., Liouville theorems on some indefinite equations, Proc. roy. soc. Edinburgh sect. A, 129, 649-661, (1999) · Zbl 0958.35056
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