We study positive Liouville theorems and the asymptotic behavior of positive solutions of p-Laplacian type elliptic equations of the form -Δp(u) + V|u|p-2 u = 0 in X, where X is a domain in ℝd, d ≥ 2 and 1 < p < ∞. We assume that the potential V has a Fuchsian type singularity at a point ζ, where either ζ = ∞ and X is a truncated C2-cone, or ζ = 0 and ζ is either an isolated point of ∂X or belongs to a C2-portion of ∂X.
Martin Fraas 1 ; Yehuda Pinchover 1
@article{CML_2011__3_2_291_0,
author = {Martin Fraas and Yehuda Pinchover},
title = {Positive {Liouville} theorems and asymptotic behavior for $p$-laplacian type elliptic equations with a {Fuchsian} potential},
journal = {Confluentes Mathematici},
pages = {291--323},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {2},
year = {2011},
doi = {10.1142/S1793744211000321},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000321/}
}
TY - JOUR AU - Martin Fraas AU - Yehuda Pinchover TI - Positive Liouville theorems and asymptotic behavior for $p$-laplacian type elliptic equations with a Fuchsian potential JO - Confluentes Mathematici PY - 2011 SP - 291 EP - 323 VL - 3 IS - 2 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000321/ DO - 10.1142/S1793744211000321 LA - en ID - CML_2011__3_2_291_0 ER -
%0 Journal Article %A Martin Fraas %A Yehuda Pinchover %T Positive Liouville theorems and asymptotic behavior for $p$-laplacian type elliptic equations with a Fuchsian potential %J Confluentes Mathematici %D 2011 %P 291-323 %V 3 %N 2 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000321/ %R 10.1142/S1793744211000321 %G en %F CML_2011__3_2_291_0
Martin Fraas; Yehuda Pinchover. Positive Liouville theorems and asymptotic behavior for $p$-laplacian type elliptic equations with a Fuchsian potential. Confluentes Mathematici, Tome 3 (2011) no. 2, pp. 291-323. doi : 10.1142/S1793744211000321. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000321/
[1] S. Agmon, On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in Methods of Functional Analysis and Theory of Elliptic Equations (Liguori, 1982), pp. 19–52.
[2] G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved Lp Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004) 2169–2196.
[3] G. Barles, Remarks on uniqueness results of the first eigenvalue of the p-Laplacian, Ann. Fac. Sci. Toulouse Math. 9 (1988) 65–75.
[4] M. F. Bidaut-Véron, R. Borghol and L. Véron, Boundary Harnack inequality and a priori estimates of singular solutions of quasilinear elliptic equations, Calc. Var. Partial Differential Equations 27 (2006) 159–177.
[5] I. Birindelli and F. Demengel, Some Liouville theorems for the p-Laplacian, in Proc. of the 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, (electronic), Electron. J. Differ. Equ. Conf. 8 (2002) 35–46.
[6] R. Borghol and L. Véron, Boundary singularities of N-harmonic functions, Comm. Partial Differential Equations 32 (2007) 1001–1015.
[7] M. Cuesta and P. Tak´aˇc, A strong comparison principle for positive solutions of degenerate elliptic equations, Differential Integral Equations 13 (2000) 721–746.
[8] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré. Anal. Non Linéaire 15 (1998) 493–516.
[9] L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison prin- ciples and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations 25 (2006) 139–159.
[10] V. De Cicco and M. A. Vivaldi, A Liouville type theorem for weighted elliptic equations, Adv. Math. Sci. Appl. 9 (1999) 183–207.
[11] E. DiBenedetto, C1+α local regularity of weak solutions of degenerate elliptic equa- tions, Nonlinear Anal. 7 (1983) 827–850.
[12] A. Farina, Liouville-type theorems for elliptic problems, in Handbook of Differen- tial Equations: Stationary Partial Differential Equations, Vol. IV (Elsevier/North- Holland, 2007), pp. 61–116.
[13] D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Local “superlinearity” and “sublin- earity” for the p-Laplacian, J. Funct. Anal. 257 (2009) 721–752.
[14] M. Fraas and Y. Pinchover, Isolated singularities of positive solutions of p-Laplacian type equations in Rd , arXiv:1008.3873.
[15] J. Garc´ıa-Meli´an and J. Sabina de Lis, Maximum and comparison principles for oper- ators involving the p-Laplacian, J. Math. Anal. Appl. 218 (1998) 49–65.
[16] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, unabridged republication of the 1993 original (Dover, 2006).
[17] S. Kichenassamy and L. Véron, Singular solutions of the p-Laplace equation, Math. Ann. 275 (1986) 599–615; 277 (1987) 352.
[18] M. R. Lancia and M. V. Marchi, Liouville theorems for Fuchsian-type operators on the Heisenberg group, Z. Anal. Anwendungen 16 (1997) 653–668.
[19] J. L. Lewis, Applications of boundary Harnack inequalities for p-harmonic functions and related topics, preprint 2009, http://www.ms.uky.edu/∼john/itcnof.pdf.
[20] J. L. Lewis, N. Lundström and K. Nyström, Boundary Harnack inequalities for oper- ators of p-Laplace type in Reifenberg flat domains, in Perspectives in Partial Differ- ential Equations, Harmonic Analysis and Applications, Proc. Sympos. Pure Math., Vol. 79 (Amer. Math. Soc., 2008), pp. 229–266.
[21] G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988) 1203–1219.
[22] P. Lindqvist, Notes on the p-Laplace equation, Report. University of Jyväskylä Department of Mathematics and Statistics, 102, University of Jyväskylä, Jyväskylä,
[23] http://www.math.ntnu.no/∼lqvist/p-laplace.pdf.
[24] V. Liskevich, S. Lyakhova and V. Moroz, Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains, J. Differential Equations 232 (2007) 212–252.
[25] Y. Pinchover, On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators, Ann. Inst. Henri Poincaré Anal. Non Linéaire 11 (1994) 313–341.
[26] Y. Pinchover and K. Tintarev, Ground state alternative for p-Laplacian with potential term, Calc. Var. Partial Differential Equations 28 (2007) 179–201.
[27] Y. Pinchover and K. Tintarev, On positive solutions of minimal growth for singular p-Laplacian with potential term, Adv. Nonlinear Stud. 8 (2008) 213–234.
[28] Y. Pinchover and K. Tintarev, On positive solutions of p-Laplacian-type equations, in Analysis, Partial Differential Equations and Applications — The Vladimir Maz’ya Anniversary Volume, Operator Theory: Advances and Applications, Vol. 193, eds. A. Cialdea et al. (Birkäuser, 2009), pp. 245–268.
[29] A. Poliakovsky and I. Shafrir, Uniqueness of positive solutions for singular problems involving the p-Laplacian, Proc. Amer. Math. Soc. 133 (2005) 2549–2557.
[30] A. Porretta and L. Véron, Separable p-harmonic functions in a cone and related quasilinear equations on manifolds, J. Eur. Math. Soc. (JEMS ) 11 (2009) 1285–1305.
[31] P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, Vol. 73 (Birkhäuser, 2007).
[32] J. Serrin, Isolated singularities of solutions of quasi-linear equations, Acta Math. 113 (1965) 219–240.
[33] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8 (1983) 773–817.
[34] L. Véron, Singularities of Solutions of Second Order Quasilinear Equations, Pitman Research Notes in Mathematics Series, Vol. 353 (Longman, 1996).
Cité par Sources :