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References

Published online by Cambridge University Press:  05 February 2016

Marius Ghergu  and
Steven D. Taliaferro
Marius Ghergu
Affiliation:
University College Dublin
Steven D. Taliaferro
Affiliation:
Texas A & M University
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Isolated Singularities in Partial Differential Inequalities , pp. 341 - 347
Publisher: Cambridge University Press
Print publication year: 2016

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References

[1] Andreucci, D. and Di Benedetto, E. 1991. On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18, 363–441.Google Scholar
[2] Andreucci, D., Herrero, M.A., and Velázquez, J.J.L. 1997. Liouville theorems and blow up behaviour in semilinear reaction diffusion systems. Ann. Inst. H. Poincaré Anal. Non Linéaire, 14, 1–53.CrossRefGoogle Scholar
[3] Armstrong, S. and Sirakov, B. 2011. Nonexistence of positive supersolutions of elliptic equations via the maximum principle. Commun. Part. Diff. Eq., 36, 2011–2047.CrossRefGoogle Scholar
[4] Baras, P. and Pierre, M. 1984. Singularités eliminables pour des équations semi-linéaires.Ann. Inst. Fourier (Grenoble), 34, 185–206.CrossRefGoogle Scholar
[5] Bidaut-Véron, M.-F. 1998. Initial blow-up for the solutions of a semilinear parabolic equation with source term. Équations aux dérivées partielles et applications, Gauthier- Villars, Ed. Sci. Méd. Elsevier, Paris, 189–198.Google Scholar
[6] Bidaut-Véron, M.-F. and Grillot, P. 1999. Singularities in elliptic systems with absorption terms. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28, 229–271.Google Scholar
[7] Bidaut-Véron, M.-F. and Pohozaev, S.I. 2001. Nonexistence results and estimates for some quasilinear elliptic problems. J. Analyse Math., 84, 1–49.CrossRefGoogle Scholar
[8] Bidaut-Véron, M.-F., Ponce, A.C., and Véron, L. 2011. Boundary singularities of positive solutions of some nonlinear elliptic equations. Calc. Var. Partial Differential Equations, 40, 183–221.CrossRefGoogle Scholar
[9] Brezis, H. 1992. Uniform estimates for solutions of −Δu = V (x)up, Partial differential equations and related subjects (Trento, 1990), Pitman Res. Notes Math. Ser., 269, Longman Sci. Tech., Harlow, 38–52.Google Scholar
[10] Brezis, H., Dupaigne, L., and Tesei, A. 2005. On a semilinear elliptic equation with inverse-square potential. Selecta Math. (N.S.), 11, 1–7.CrossRefGoogle Scholar
[11] Brezis, H. and Friedman, A. 1983. Nonlinear parabolic equations involving measures as initial conditions. J. Math. Pures Appl., 62, 73–97.Google Scholar
[12] Brezis, H. and Lions, P.-L. 1981. A note on isolated singularities for linear elliptic equations, Mathematical Analysis and Applications, Part A, Adv. inMath. Suppl. Stud., vol. 7a, Academic Press, New York, London, 263–266.Google Scholar
[13] Brezis, H., Peletier, L.A., and Terman, D. 1986. A very singular solution of the heat equation with absorption. Arch. Rational Mech. Anal., 95, 185–209.CrossRefGoogle Scholar
[14] Brezis, H. and Merle, F. 1991. Uniform estimates and blow-up behavior for solutions of −Δu = V (x)eu in two dimensions. Commun. Part. Diff. Eq., 16, 1223–1253.CrossRefGoogle Scholar
[15] Brezis, H. and Véron, L. 1980. Removable singularities of some nonlinear equations. Arch. Rational Mech. Anal., 75, 1–6.CrossRefGoogle Scholar
[16] Caffarelli, L.A., Gidas, B., and Spruck, J. 1989. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math., 42, 271–297.CrossRefGoogle Scholar
[17] Caristi, G. 2000. Existence and nonexistence of global solutions of degenerate and singular parabolic systems. Abstr. Appl. Anal., 5, 265–284.CrossRefGoogle Scholar
[18] Caristi, G. 2001. Nonexistence theorems for systems of quasilinear parabolic inequalities. Adv. Nonlinear Stud. 1, 143–158.CrossRefGoogle Scholar
[19] Caristi, G., D'A mbrosio, L., and Mitidieri, E. 2008. Representation formulae for solutions to some classes of higher order systems and related Liouville theorems. Milan J. Math., 76, 27–67.CrossRefGoogle Scholar
[20] Caristi, G., D'A mbrosio, L., and Mitidieri, E. 2008. Liouville theorems for some nonlinear inequalities. Proc. Steklov Inst. Math., 260, 90–111.CrossRefGoogle Scholar
[21] Chao, J. and Li, C. 2006. Quantitative analysis of some system of integral equations. Calc. Var. Partial Differential Equations, 26, 447–457.Google Scholar
[22] Chao, J. and Li, C. 2006. Symmetry of solutions to some systems of integral equations. Proc. Amer. Math. Soc., 134, 1661–1670.Google Scholar
[23] Chen, W. and Li, C. 2009. An integral system and the Lane-Emden conjecture. Discrete Contin. Dyn. Syst. 24, 1167–1184.Google Scholar
[24] Chen, W. and Li, C. 2013. Super polyharmonic property of solutions for PDE systems and its applications. Commun. Pure Appl. Anal., 12, 2497–2514.Google Scholar
[25] Chen, W., Li, C., and Ou, B. 2005. Classification of solutions for a system of integral equations. Commun. Part. Diff. Eq., 30, 59–65.CrossRefGoogle Scholar
[26] Chen, W., Li, C., and Ou, B. 2006. Classification of solutions for an integral equation. Comm. Pure Appl. Math., 59, 330–343.Google Scholar
[27] Chen, C.-C. and Lin, C.-S. 1997. Estimates of the conformal scalar curvature equation via the method of moving planes. Comm. Pure Appl. Math., 50, 971–1017.3.0.CO;2-D>CrossRefGoogle Scholar
[28] Chen, H. and Lü, Z. 2014. The properties of positive solutions to an integral system involving Wolff potential. Discrete Contin. Dyn. Syst., 34, 1879–1904.Google Scholar
[29] Chen, X.Y., Matano, H., and Véron, L. 1989. Anisotropic singularities of nonlinear elliptic equations in R2. J. Funct. Anal., 83, 50–97.CrossRefGoogle Scholar
[30] Cîrstea, F.-C. 2014. A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials. Mem. Amer. Math. Soc., 227.Google Scholar
[31] Davies, E.B. 1989. Heat Kernels and Spectral Theory, Cambridge University Press.CrossRefGoogle Scholar
[32] D'A mbrosio, L. and Mitidieri, E. 2010. A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities. Adv. Math., 224, 967– 1020.Google Scholar
[33] D'A mbrosio, L., Mitidieri, E., and Pohozaev, S.I. 2006. Representation formulae and inequalities for solutions of a class of second order partial differential equations. Trans. Amer. Math. Soc., 358, 893–910.Google Scholar
[34] del Pino, M., Musso, M., and Pacard, F. 2007. Boundary singularities for weak solutions of semilinear elliptic problems. J. Funct. Anal., 253, 241–272.CrossRefGoogle Scholar
[35] Devreese, J.T. and Alexandrov, A.S. 2010. Advances in Polaron Physics, Springer Series in Solid-State Sciences, vol. 159, Springer.Google Scholar
[36] Fall, M.M. 2012. On the Hardy-Poincaré inequality with boundary singularities. Commun. Contemp. Math., 14, 1250019, 13 pp.CrossRefGoogle Scholar
[37] Fall, M.M. 2013. Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential. J. Funct. Anal., 264, 661–690.CrossRefGoogle Scholar
[38] Fall, M.M. and Musina, R. 2012. Hardy-Poincaré inequalities with boundary singularities. Proc. Roy. Soc. Edinburgh Sect. A, 142, 769–786.CrossRefGoogle Scholar
[39] Farina, A. and Serrin, J. 2011. Entire solutions of completely coercive quasilinear elliptic equations I. J. Differential Equations, 250, 4367–4408.Google Scholar
[40] Farina, A. and Serrin, J. 2011. Entire solutions of completely coercive quasilinear elliptic equations II. J. Differential Equations, 250, 4409–4436.Google Scholar
[41] Filippucci, R. 2011. Nonexistence of nonnegative nontrivial solutions of elliptic systems of the divergence type. J. Differential Equations, 250, 572–595.CrossRefGoogle Scholar
[42] Filippucci, R. 2013. Quasilinear elliptic systems in RN with multipower forcing terms depending on the gradient. J. Differential Equations, 255, 1839–1866.CrossRefGoogle Scholar
[43] Filippucci, R., Pucci, P., and Rigoli, M. 2009. On weak solutions of nonlinear weighted p-Laplacian elliptic inequalities. Nonlinear Anal., 70, 3008–3019.Google Scholar
[44] Filippucci, R., Pucci, P., and Rigoli, M. 2010. Nonlinear weighted p-Laplacian elliptic inequalities with gradient terms. Commun. Contemp. Math., 12, 501–535.CrossRefGoogle Scholar
[45] Fowler, R.H. 1931. Further studies on Emden's and similar differential equations. Quart. J. Math. Oxford Ser., 2, 259–288.Google Scholar
[46] Futamura, T., Kishi, K., and Mizuta, Y. 2003. Removability of sets for sub-polyharmonic functions. Hiroshima Math. J., 33, 31–42.Google Scholar
[47] Futamura, T. and Mizuta, Y. 2004. Isolated singularities of super-polyharmonic functions. Hokkaido Math. J., 33, 675–695.CrossRefGoogle Scholar
[48] Futamura, T., Mizuta, Y., and Ohno, T. 2014. Riesz decomposition for superpolyharmonic functions in the punctured unit ball. Studia Sci. Math. Hungar., 51, 1–16.Google Scholar
[49] Ghergu, M., Liskevich, V., and Sobol, Z. 2014. Singular solutions for second-order nondivergence type elliptic inequalities in punctured balls. J. Analyse Math., 123, 251–279.CrossRefGoogle Scholar
[50] Ghergu, M., Moradifam, A., and Taliaferro, S.D. 2011. Isolated singularities of polyharmonic inequalities. J. Funct. Anal., 261, 660–680.CrossRefGoogle Scholar
[51] Ghergu, M. and Taliaferro, S.D. 2015. Nonexistence of positive supersolutions of nonlinear biharmonic equations without the maximum principle. Commun. Part. Diff. Eq., 6, 1029–1069.Google Scholar
[52] Ghergu, M. and Taliaferro, S.D. 2015. Asymptotic behavior at isolated singularities for solutions of nonlocal semilinear elliptic systems of inequalities. Calc. Var. Partial Differential Equations, 54, 1243–1273.CrossRefGoogle Scholar
[53] Ghergu, M. and Taliaferro, S.D. 2015. Pointwise bounds and blow-up for systems of semilinear parabolic inequalities and nonlinear heat potential estimates. arXiv:1504.04841 [math.AP].
[54] Ghergu, M., Taliaferro, S.D., and Verbitsky, I.E. 2015. Pointwise bounds and blow-up for systems of semilinear elliptic inequalities at an isolated singularity via nonlinear potential estimates. arXiv:1402.0113 [math.AP].
[55] Gidas, B. and Spruck, J. 1981. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math., 34, 525–598.CrossRefGoogle Scholar
[56] Giga, Y. and Kohn, R.V. 1987. Characterizing blowup using similarity variables. Indiana Univ. Math. J., 36, 1–40.CrossRefGoogle Scholar
[57] Giga, Y., Matsui, S., and Sasayama, S. 2004. Blow up rate for semilinear heat equations with subcritical nonlinearity. Indiana Univ. Math. J., 53, 483–514.CrossRefGoogle Scholar
[58] Gilbarg, D. and Serrin, J. 1955/56. On isolated singularities of solutions of second order elliptic differential equations. J. Analyse Math., 4, 309–340.Google Scholar
[59] Gilbarg, D. and Trudinger, N. 1983. Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin.CrossRefGoogle Scholar
[60] Hedberg, L.I. 1972. On certain convolution inequalities. Proc. Amer. Math. Soc., 36, 505–510.CrossRefGoogle Scholar
[61] Herrero, M.A. and Velázquez, J.J.L. 1994. Explosion de solutions d’équations paraboliques semilinéaires supercritiques. C. R. Acad. Sci. Paris Sér. I Math. 319, 141–145.Google Scholar
[62] Hirata, K. 2014. Removable singularities of semilinear parabolic equations. Proc. Amer. Math. Soc., 142, 157–171.Google Scholar
[63] Hsu, S.Y. 2010. Removable singularities of semilinear parabolic equations. Adv. Differential Equations, 15, 137–158.Google Scholar
[64] Hui, K.M. 1992. A Fatou theorem for the solution of the heat equation at the corner points of a cylinder. Trans. Amer. Math. Soc., 333, 607–642.CrossRefGoogle Scholar
[65] Hui, K.M. 2010. Another proof for the removable singularities of the heat equation. Proc. Amer. Math. Soc., 138, 2397–2402.CrossRefGoogle Scholar
[66] Hardy, G.H., Littlewood, J.E., and Polya, G. 1934. Inequalities, Cambridge University Press.Google Scholar
[67] Jin, C. and Li, C. 2006. Quantitative analysis of some system of integral equations. Calc. Var. Partial Differential Equations, 26, 447–457.CrossRefGoogle Scholar
[68] Jones, K.R.W. 1995. Newtonian quantum gravity. Aust. J. Phys., 48, 1055–1081.CrossRefGoogle Scholar
[69] Kato, T. 1972. Schrödinger operators with singular potentials. Israel J. Math., 13, 135– 148.CrossRefGoogle Scholar
[70] Kavian, O. 1987. Remarks on the large time behaviour of a nonlinear diffusion equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 4, 423–452.CrossRefGoogle Scholar
[71] Kondratiev, V., Liskevich, V., and Sobol, Z. 2003. Second-order semilinear elliptic inequalities in exterior domains. J. Differential Equations, 187, 429–455.CrossRefGoogle Scholar
[72] Kondratiev, V., Liskevich, V., and Sobol, Z. 2008. Positive solutions to semi-linear and quasilinear second-order elliptic equations on unbounded domains, in:M. Chipot (Ed.), Handbook of Differential Equation: Stationary Partial Differential Equations, vol. 6, Elsevier, 177–268.Google Scholar
[73] Kondratiev, V., Liskevich, V., and Sobol, Z. 2009. Positive supersolutions to semi-linear second-order non-divergence type elliptic equations in exterior domains. Trans. Amer. Math. Soc., 361, 697–713.Google Scholar
[74] Kondratiev, V., Liskevich, V., Sobol, Z., and Us, A. 2004. 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., 69, 107–127.CrossRefGoogle Scholar
[75] Korevaar, N., Mazzeo, R., Pacard, F., and Schoen, R. 1999. Refined asymptotics for constant scalar curvature metrics with isolated singularities. Invent. Math., 135, 233–272.CrossRefGoogle Scholar
[76] Ladyzhenskaya, O.A., Solonnikov, V.A., and Ural'tseva, N.N. 1968. Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc, Providence, R.I.CrossRefGoogle Scholar
[77] Leung, M.-C. 2003. Blow-up of nonlinear elliptic equations in Rn with critical exponent. Math. Ann., 327, 723–744.Google Scholar
[78] Lei, Y. 2013. On the regularity of positive solutions of a class of Choquard type equations. Math. Z., 273, 883–905.CrossRefGoogle Scholar
[79] Lei, Y. 2013. Qualitative analysis for the static Hartree-type equations. SIAM J. Math. Anal., 45, 388–406.CrossRefGoogle Scholar
[80] Lei, Y. and Li, C. 2012. Integrability and asymptotics of positive solutions of a λ -Laplace system. J. Differential Equations, 252, 2739–2758.CrossRefGoogle Scholar
[81] Lei, Y., Li, C., and Ma, C. 2012. Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations. Calc. Var. Partial Differential Equations, 45, 43–61.CrossRefGoogle Scholar
[82] Lei, Y. and Ma, C. 2012. Radial symmetry and decay rates of positive solutions of a Wolff type integral system. Proc. Amer. Math. Soc., 140, 541–551.CrossRefGoogle Scholar
[83] Li, C. 1996. Local asymptotic symmetry of singular solutions to nonlinear elliptic equations. Invent. Math., 123, 221–231.CrossRefGoogle Scholar
[84] Lieb, E.H. 1976/77. Existence and uniqueness of theminimizing solution of Choquard's nonlinear equation. Studies in Appl. Math., 57, 93–105.Google Scholar
[85] Lieberman, G. 1996. Second Order Parabolic Differential Equations, World Scientific, Singapore.CrossRefGoogle Scholar
[86] Lin, C.-S. 2000. Estimates of the scalar curvature equation via the method of moving planes III. Comm. Pure Appl. Math., 53, 611–646.3.0.CO;2-N>CrossRefGoogle Scholar
[87] Liskevich, V., Lyakova, S., and Moroz, V. 2006. Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains. Adv. Differential Equations, 11, 361–398.Google Scholar
[88] Liskevich, V., Shishkov, I., and Sobol, Z. 2012. Singular solutions to the heat equations with nonlinear absorption and Hardy potentials. Commun. Contem. Math., 14, 1250013.CrossRefGoogle Scholar
[89] Ma, C., Chen, W., and Li, C. 2011. Regularity of solutions for an integral system of Wolff type. Adv. in Math., 226, 2676–2699.CrossRefGoogle Scholar
[90] Matano, H. and Merle, F. 2004. On nonexistence of type II blowup for a supercritical nonlinear heat equation. Comm. Pure Appl. Math., 57, 1494–1541.CrossRefGoogle Scholar
[91] Maz'ya, V.G. 2011. Sobolev Spaces, with Applications to Elliptic Partial Differential Equations, 2nd augmented edition. Grundlehren der Math. Wissenschaften 342, Springer, Berlin.Google Scholar
[92] Maz'ya, V.G. and Havin, V.P. 1972. Nonlinear potential theory. Usp. Mat. Nauk, 27, 67–138 (in Russian). English translation: Russ. Math. Surv., 27, 71–148.Google Scholar
[93] Merle, F. and Zaag, H. 1998. Optimal estimates for blowup rate and behavior for nonlinear heat equations. Comm. Pure Appl. Math., 51, 139–196.3.0.CO;2-C>CrossRefGoogle Scholar
[94] Meyers, N. and Serrin, J. 1960. The exterior Dirichlet problem for second order elliptic partial differential equations. J. Math. Mech., 9, 513–538.Google Scholar
[95] Mitidieri, E. 1993. A Rellich type identity and applications. Commun. Part. Diff. Eq., 18, 125–151.CrossRefGoogle Scholar
[96] Mitidieri, E. 1996. Nonexistence of positive solutions of semilinear elliptic systems in RN. Differential Integral Equations, 9, 465–479.Google Scholar
[97] Mitidieri, E. and Pohozaev, S.I. 1998. Nonexistence of global positive solutions to quasilinear elliptic inequalities. Dokl. Akad. Nauk, 359, 456–460.Google Scholar
[98] Mitidieri, E. and Pohozaev, S.I. 1999. Nonexistence of positive solutions to quasilinear elliptic problems in RN. Proc. Steklov Institute of Math., 227, 192–222.Google Scholar
[99] Mitidieri, E. and Pohozaev, S.I. 2001. A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Proc. Steklov Math. Inst., 234, 1–362.Google Scholar
[100] Mitidieri, E. and Pohozaev, S.I. 2004. Towards a unified approach to nonexistence of solutions for a class of differential inequalities. Milan J. Math., 72, 129–162.CrossRefGoogle Scholar
[101] Mizoguchi, N. 2004. Type-II blowup for a semilinear heat equation. Adv. Differential Equations, 9, 1279–1316.Google Scholar
[102] Moroz, I.M., Penrose, R., and Tod, P. 1998. Spherically-symmetric solutions of the Schroödinger-Newton equations. Classical Quantum Gravity, 15, 2733–2742.CrossRefGoogle Scholar
[103] Moutoussamy, I. and Véron, L. 1989. Source type positive solutions of nonlinear parabolic inequalities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16, 527–555.Google Scholar
[104] Oswald, L. 1988. Isolated positive singularities for a nonlinear heat equation. Houston J. Math., 14, 543–572.Google Scholar
[105] Pekar, S. 1954. Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin.Google Scholar
[106] Pinchover, Y. 1994. On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators. Ann. Inst. H. Poincaré Anal. Non Linéaire, 11, 313–341.CrossRefGoogle Scholar
[107] Poláčik, P., Quittner, P., and Souplet, Ph. 2007. Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II: Parabolic equations, Indiana Univ. Math. J., 56, 879–908.CrossRefGoogle Scholar
[108] Poláčik, P. and Yanagida, E. 2003. On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann., 327, 745–771.Google Scholar
[109] Quittner, P. and Souplet, Ph. 2007. Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhauser.Google Scholar
[110] Quittner, P., Souplet, Ph., and Winkler, M. 2004. Initial blow-up rates and universal bounds for nonlinear heat equations. J. Differential Equations, 196, 316–339.CrossRefGoogle Scholar
[111] Richard, Y. and Véron, L. 1989. Isotropic singularities of solutions of nonlinear elliptic inequalities. Ann. Inst. H. Poincaré, 6, 37–72.Google Scholar
[112] Sato, S. and Yanagida, E. 2009. Solutions with moving singularities for a semilinear parabolic equation. J. Differential Equations, 246, 724–748.CrossRefGoogle Scholar
[113] Schoen, R. 1991. On the number of constant scalar curvature metrics in a conformal class, in Differential Geometry: A Symposium in Honor of Manfredo Do Carmo, H.B, Lawson and K, Tenenblat (Eds.), 311–320, Wiley, New York.Google Scholar
[114] Serrin, J. 1964. Removable singularities of solutions of elliptic equations. Arch. Rational Mech. Anal., 17, 67–76.CrossRefGoogle Scholar
[115] Serrin, J. λ. Removable singularities of solutions of elliptic equations II. Arch. Rational Mech. Anal., 20, 163–169.
[116] Serrin, J. 1965. Isolated singularities of solutions of quasilinear equations. Acta Math., 113, 219–240.CrossRefGoogle Scholar
[117] Serrin, J. and Zou, H. 2002. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math., 189, 79–142.CrossRefGoogle Scholar
[118] Shishkov, A. and Véron, L. 2008. Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption. Calc. Var. PartialDifferential Equations, 33, 343–375.Google Scholar
[119] Souplet, Ph. 2009. The proof of the Lane-Emden conjecture in four space dimensions. Adv. Math., 221, 1409–1427.CrossRefGoogle Scholar
[120] Stein, E.M. 1971. Singular Integrals and Differentiability Properties of Functions Properties of Functions, Princeton Univ. Press.CrossRefGoogle Scholar
[121] Sun, Y. and Lei, Y. 2012. Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials. J. Funct. Anal., 263, 3857–3882.CrossRefGoogle Scholar
[122] Takahashi, J. and Yanagida, E. 2015. Time-dependent singularities in the heat equation. Comm. Pure Appl. Anal., 3, 969–979.Google Scholar
[123] Taliaferro, S.D. 1978. Asymptotic behavior of solutions of y”= φ(t)yλ.J. Math. Anal. Appl., 66, 95–134.CrossRefGoogle Scholar
[124] Taliaferro, S.D. 2001. Isolated singularities of nonlinear elliptic inequalities. Indiana Univ. Math. J., 50, 1885–1897.CrossRefGoogle Scholar
[125] Taliaferro, S.D. 2001. On the growth of superharmonic functions near an isolated singularity II, Commun. Part. Diff. Eq., 26, 1003–1026.CrossRefGoogle Scholar
[126] Taliaferro, S.D. 2002. Local behavior and global existence of positive solutions of auλ = -Δu = uλ. Ann. Inst. H. Poincare Anal. Non Lineaire, 19, 889–901.CrossRefGoogle Scholar
[127] Taliaferro, S.D. 2006. Isolated singularities of nonlinear elliptic inequalities. II. Asymptotic behavior of solutions. Indiana Univ. Math. J., 55, 1791–1812.CrossRefGoogle Scholar
[128] Taliaferro, S.D. 2007. Isolated singularities of nonlinear parabolic inequalities. Math. Ann., 338, 555–586.CrossRefGoogle Scholar
[129] Taliaferro, S.D. 2009. Blow-up of solutions of nonlinear parabolic inequalities. Trans. Amer. Math. Soc., 361, 3289–3302.CrossRefGoogle Scholar
[130] Taliaferro, S.D. 2011. Initial blow-up of solutions of semilinear parabolic inequalities. J. Differential Equations, 250, 892–928.CrossRefGoogle Scholar
[131] Taliaferro, S.D. 2013. Pointwise bounds and blow-up for nonlinear polyharmonic inequalities. Ann. Inst. H. Poincaré Anal. Non Linéaire, 30, 1069–1096.CrossRefGoogle Scholar
[132] Taliaferro, S.D. and Zhang, L. 2003. Arbitrarily large solutions of the conformal scalar curvature problem at an isolated singularity. Proc Amer. Math. Soc., 131, 2895–2902.CrossRefGoogle Scholar
[133] Terracini, S. 1996. On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differential Equations, 1, 241–264.Google Scholar
[134] Véron, L. 1981. Singular solutions of some nonlinear elliptic equations. Nonlinear Anal. TMA 5, 225–241.CrossRefGoogle Scholar
[135] Véron, L. 1996. Singularities of Solutions of Second Order Quasilinear Equations. Longman.Google Scholar
[136] Villavert, J. 2015. Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality. J. Differential Equations, 258, 1685–1714.CrossRefGoogle Scholar
[137] Villavert, J. 2015. A characterization of fast decaying solutions for quasilinear andWolff type systems with singular coefficients. J. Math. Anal. Appl., 424, 1348–1373.CrossRefGoogle Scholar
[138] Watson, N.A. 1976. Green functions, potentials, and the Dirichlet problem for the heat equation. Proc. London Math. Soc., 33, 251–298.Google Scholar
[139] Zhang, F. 1999. Matrix Theory. Basic Results and Techniques, Springer-Verlag.CrossRefGoogle Scholar
[140] Zhang, L. 2002. Refined asymptotic estimates for conformal scalar curvature equation via moving sphere method. J. Funct. Anal., 192, 491–516.CrossRefGoogle Scholar
[141] Zhang, Q.S. 2002. The boundary behavior of heat kernels of Dirichlet Laplacians. J. Differential Equations, 182, 416–430.CrossRefGoogle Scholar
[142] Yu, X. 2013. Liouville type theorems for integral equations and integral systems. Calc. Var. Partial Differential Equations, 46, 75–95.CrossRefGoogle Scholar

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  • References
  • Marius Ghergu, University College Dublin, Steven D. Taliaferro, Texas A & M University
  • Book: Isolated Singularities in Partial Differential Inequalities
  • Online publication: 05 February 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781316481363.016
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  • References
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  • Book: Isolated Singularities in Partial Differential Inequalities
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  • Chapter DOI: https://doi.org/10.1017/CBO9781316481363.016
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  • References
  • Marius Ghergu, University College Dublin, Steven D. Taliaferro, Texas A & M University
  • Book: Isolated Singularities in Partial Differential Inequalities
  • Online publication: 05 February 2016
  • Chapter DOI: https://doi.org/10.1017/CBO9781316481363.016
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