Uniqueness of Radial Solutions for the Fractional Laplacian
We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation (−Δ)su+Vu=0 in ℝN has at m...
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| Published in | Communications on pure and applied mathematics Vol. 69; no. 9; pp. 1671 - 1726 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Blackwell Publishing Ltd
01.09.2016
John Wiley and Sons, Limited |
| Subjects | |
| Online Access | Get full text |
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| Abstract | We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation
(−Δ)su+Vu=0 in ℝN
has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (−Δ)s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space ℝ+N+1, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation
−Δ)sQ+Q−|Q|αQ=0 in ℝN
for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc. |
|---|---|
| AbstractList | We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)
s
with
s
∊ (0,1) for any space dimensions
N
≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation
urn:x-wiley:00103640:media:cpa21591:cpa21591-math-0001
has at most one radial and bounded solution vanishing at infinity, provided that the potential
V
is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator
H
= (−Δ)
s
+
V
are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space
, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation
urn:x-wiley:00103640:media:cpa21591:cpa21591-math-0003
for arbitrary space dimensions
N
≥ 1 and all admissible exponents
α
> 0. This generalizes the nondegeneracy and uniqueness result for dimension
N
= 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc. We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (...)^sup s^ with s... (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabre and Sire , we show that the linear equation ... has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrodinger operator H = (-...)^sup s^ + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space ..., we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation ... for arbitrary space dimensions N ≥ 1 and all admissible exponents a > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland. (ProQuest: ... denotes formulae/symbols omitted.) We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)s with s ∊ (0,1) for any space dimensions N ≥ 1. By extending a monotonicity formula found by Cabré and Sire , we show that the linear equation (−Δ)su+Vu=0 in ℝN has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator H = (−Δ)s + V are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half‐space ℝ+N+1, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation −Δ)sQ+Q−|Q|αQ=0 in ℝN for arbitrary space dimensions N ≥ 1 and all admissible exponents α > 0. This generalizes the nondegeneracy and uniqueness result for dimension N = 1 recently obtained by the first two authors and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc. |
| Author | Lenzmann, Enno Frank, Rupert L. Silvestre, Luis |
| Author_xml | – sequence: 1 givenname: Rupert L. surname: Frank fullname: Frank, Rupert L. email: rlfrank@caltech.edu organization: Caltech Mathematics, 253-37, CA, 91125, Pasadena, USA – sequence: 2 givenname: Enno surname: Lenzmann fullname: Lenzmann, Enno email: enno.lenzmann@unibas.ch organization: University of Basel, Department of Mathematics and Computer Science, Spiegelgasse 1, CH-4051, Basel, Switzerland – sequence: 3 givenname: Luis surname: Silvestre fullname: Silvestre, Luis email: luis@math.uchicago.edu organization: University of Chicago, Mathematics Department, 5734 S. University Ave. Office Ry 360-E, IL, 60637, Chicago, USA |
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| Copyright | 2016 Wiley Periodicals, Inc. Copyright John Wiley and Sons, Limited Sep 2016 |
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| References | Albert, J. P.; Bona, J. L.; Saut, J.-C. Model equations for waves in stratified fluids. Proc. Roy. Soc. London Ser. A 453 (1997), no. 1961, 1233-1260. doi: 10.1098/rspa.1997.0068 Abdelouhab, L.; Bona, J. L.; Felland, M.; Saut, J.-C. Nonlocal models for nonlinear, dispersive waves. Phys. D 40 (1989), no. 3, 360-392. doi: 10.1016/0167-2789(89)90050-X Amick, C. J.; Toland, J. F. Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane. Acta Math. 167 (1991), no. 1-2, 107-126. doi: 10.1007/BF02392447 Blumenthal, R. M.; Getoor, R. K. Some theorems on stable processes. Trans. Amer. Math. Soc. 95 (1960), 263-273. doi: 10.2307/1993291 Weinstein, M. I. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), no. 3, 472-491. doi: 10.1137/0516034 Graham, C. R.; Zworski, M. Scattering matrix in conformal geometry. Invent. Math. 152 (2003), no. 1, 89-118. doi: 10.1007/s00222-002-0268-1 Krieger, J.; Lenzmann, E.; Raphaël, P. Nondispersive solutions to the L2-critical half-wave equation. Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61-129. doi: 10.1007/s00205-013-0620-1 Kenig, C. E.; Martel, Y.; Robbiano, L. Local well-posedness and blow-up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 6, 853-887. doi: 10.1016/j.anihpc.2011.06.005 Lieb, E. H.; Loss, M. Analysis. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, R.I., 1997. McLeod, K. Uniqueness of positive radial solutions of Δ u + f(u) = 0 in Rn. II. Trans. Amer. Math. Soc. 339 (1993), no. 2, 495-505. doi: 10.2307/2154282 Chang, S.-M.; Gustafson, S.; Nakanishi, K.; Tsai, T.-P. Spectra of linearized operators for NLS solitary waves. SIAM J. Math. Anal. 39 (2007/08), no. 4, 1070-1111. doi: 10.1137/050648389 Kwong, M. K. Uniqueness of positive solutions of Δ u-u + up = 0 in Rn. Arch. Rational Mech. Anal. 105 (1989), no. 3, 243-266. doi: 10.1007/BF00251502 Banica, V.; del Mar González, M.; Saéz, M. Some constructions for the fractional Laplacian on noncompact manifolds. Rev. Mat. Iber., forthcoming. arxiv:1212.3109 [math.DG] Weinstein, M. I. Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Comm. Partial Differential Equations 12 (1987), no. 10, 1133-1173. doi: 10.1080/03605308708820522 Bañuelos, R.; Kulczycki, T. The Cauchy process and the Steklov problem. J. Funct. Anal. 211 (2004), no. 2, 355-423. doi: 10.1016/j.jfa.2004.02.005 Fabes, E. B.; Kenig, C. E.; Serapioni, R. P. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7 (1982), no. 1, 77-116. doi: 10.1080/03605308208820218 Silvestre, L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60 (2007), no. 1, 67-112. doi: 10.1002/cpa.20153 Carmona, R.; Masters, W. C.; Simon, B. Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91 (1990), no. 1, 117-142. doi: 10.1016/0022-1236(90)90049-Q Caffarelli, L.; Silvestre, L. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. doi: 10.1080/03605300600987306 Kulczycki, T.; Kwaśnicki, M.; Mał;ecki, J.; Stos, A. Spectral properties of the Cauchy process on half-line and interval. Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 589-622. doi: 10.1112/plms/pdq010 Weinstein, M. I. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1982/83), no. 4, 567-576. Alessandrini, G.; Magnanini, R. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal. 25 (1994), no. 5, 1259-1268. doi: 10.1137/S0036141093249080 Molčanov, S. A.; Ostrovskiĭ, E. Symmetric stable processes as traces of degenerate diffusion processes. Teor. Verojatnost. i Primenen. 14 (1969), 127-130. Chang, S.-Y. A.; Gonzàlez, M. d. M. Fractional Laplacian in conformal geometry. Adv. Math. 226 (2011), no. 2, 1410-1432. doi: 10.1016/j.aim.2010.07.016 Frank, R. L.; Seiringer, R. Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255 (2008), no. 12, 3407-3430. doi: 10.1016/j.jfa.2008.05.015 Kwaśnicki, M. Eigenvalues of the fractional Laplace operator in the interval. J. Funct. Anal. 262 (2012), no. 5, 2379-2402. doi: 10.1016/j.jfa.2011.12.004 Chen, W.; Li, C.; Ou, B. Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59 (2006), no. 3, 330-343. doi: 10.1002/cpa.20116 Reed, M.; Simon, B. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978. Cabré, X.; Sire, Y. Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 1, 23-53. doi: 10.1016/j.anihpc.2013.02.001 Fall, M. M.; Valdinoci, E. Uniqueness and nondegeneracy of positive solutions of (−Δ)su + u = up in ℝN when s is close to 1. Comm. Math. Phys. 329 (2014), no. 1, 383-404. doi: 10.1007/s00220-014-1919-y Cabré, X.; Solà-Morales, J. Layer solutions in a half-space for boundary reactions. Comm. Pure Appl. Math. 58 (2005), no. 12, 1678-1732. doi: 10.1002/cpa.20093 Frank, R. L.; Lenzmann, E. Uniqueness of non-linear ground states for fractional Laplacians in ℝ. Acta Math. 210 (2013), no. 2, 261-318. doi: 10.1007/s11511-013-0095-9 Chen, W.; Li, C.; Ou, B. Qualitative properties of solutions for an integral equation. Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 347-354. Lenzmann, E. Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE 2 (2009), no. 1, 1-27. doi: 10.2140/apde.2009.2.1 Coffman, C. V. Uniqueness of the ground state solution for Δ u-u + u3 = 0 and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81-95. doi: 10.1007/BF00250684 Burchard, A.; Hajaiej, H. Rearrangement inequalities for functionals with monotone integrands. J. Funct. Anal. 233 (2006), no. 2, 561-582. doi: 10.1016/j.jfa.2005.08.010 Li, Y. Y. Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2, 153-180. Ma, L.; Zhao, L. Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195 (2010), no. 2, 455-467. doi: 10.1007/s00205-008-0208-3 Merle, F.; Raphael, P. The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. (2) 161 (2005), no. 1, 157-222. doi: 10.4007/annals.2005.161.157 1989; 40 1960; 95 2012; 262 1987; 12 2013; 209 2010; 101 1997; 453 2006; 59 1997 1994; 25 2004; 6 1969; 14 2007; 32 1972; 46 2006; 233 1982/83; 87 1978 2003; 152 2004; 211 2011; 226 1989; 105 2005; 161 1991; 167 2014; 329 1993; 339 2013; 210 1982; 7 2007; 60 2010; 195 2008; 255 2011; 28 2009; 2 1990; 91 2005; 12 1985; 16 2007/08; 39 2014; 31 2005; 58 e_1_2_1_20_1 e_1_2_1_40_1 e_1_2_1_23_1 e_1_2_1_24_1 e_1_2_1_21_1 e_1_2_1_22_1 e_1_2_1_27_1 e_1_2_1_28_1 e_1_2_1_25_1 e_1_2_1_26_1 e_1_2_1_29_1 Lieb E. H. (e_1_2_1_31_1) 1997 Reed M. (e_1_2_1_36_1) 1978 e_1_2_1_7_1 e_1_2_1_8_1 e_1_2_1_30_1 e_1_2_1_5_1 e_1_2_1_3_1 e_1_2_1_12_1 e_1_2_1_4_1 e_1_2_1_13_1 e_1_2_1_34_1 e_1_2_1_10_1 e_1_2_1_33_1 e_1_2_1_2_1 Banica V. (e_1_2_1_6_1) e_1_2_1_11_1 e_1_2_1_32_1 e_1_2_1_16_1 e_1_2_1_39_1 e_1_2_1_17_1 e_1_2_1_38_1 e_1_2_1_14_1 Molčanov S. A. (e_1_2_1_35_1) 1969; 14 e_1_2_1_37_1 e_1_2_1_15_1 e_1_2_1_9_1 e_1_2_1_18_1 e_1_2_1_19_1 |
| References_xml | – reference: Cabré, X.; Sire, Y. Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 1, 23-53. doi: 10.1016/j.anihpc.2013.02.001 – reference: Chang, S.-M.; Gustafson, S.; Nakanishi, K.; Tsai, T.-P. Spectra of linearized operators for NLS solitary waves. SIAM J. Math. Anal. 39 (2007/08), no. 4, 1070-1111. doi: 10.1137/050648389 – reference: Caffarelli, L.; Silvestre, L. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. doi: 10.1080/03605300600987306 – reference: Lieb, E. H.; Loss, M. Analysis. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, R.I., 1997. – reference: Frank, R. L.; Lenzmann, E. Uniqueness of non-linear ground states for fractional Laplacians in ℝ. Acta Math. 210 (2013), no. 2, 261-318. doi: 10.1007/s11511-013-0095-9 – reference: Weinstein, M. I. Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Comm. Partial Differential Equations 12 (1987), no. 10, 1133-1173. doi: 10.1080/03605308708820522 – reference: Banica, V.; del Mar González, M.; Saéz, M. Some constructions for the fractional Laplacian on noncompact manifolds. Rev. Mat. Iber., forthcoming. arxiv:1212.3109 [math.DG] – reference: Frank, R. L.; Seiringer, R. Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255 (2008), no. 12, 3407-3430. doi: 10.1016/j.jfa.2008.05.015 – reference: Kenig, C. E.; Martel, Y.; Robbiano, L. Local well-posedness and blow-up in the energy space for a class of L2 critical dispersion generalized Benjamin-Ono equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 6, 853-887. doi: 10.1016/j.anihpc.2011.06.005 – reference: Blumenthal, R. M.; Getoor, R. K. Some theorems on stable processes. Trans. Amer. Math. Soc. 95 (1960), 263-273. doi: 10.2307/1993291 – reference: Fall, M. M.; Valdinoci, E. Uniqueness and nondegeneracy of positive solutions of (−Δ)su + u = up in ℝN when s is close to 1. Comm. Math. Phys. 329 (2014), no. 1, 383-404. doi: 10.1007/s00220-014-1919-y – reference: Cabré, X.; Solà-Morales, J. Layer solutions in a half-space for boundary reactions. Comm. Pure Appl. Math. 58 (2005), no. 12, 1678-1732. doi: 10.1002/cpa.20093 – reference: Molčanov, S. A.; Ostrovskiĭ, E. Symmetric stable processes as traces of degenerate diffusion processes. Teor. Verojatnost. i Primenen. 14 (1969), 127-130. – reference: Burchard, A.; Hajaiej, H. Rearrangement inequalities for functionals with monotone integrands. J. Funct. Anal. 233 (2006), no. 2, 561-582. doi: 10.1016/j.jfa.2005.08.010 – reference: Alessandrini, G.; Magnanini, R. Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions. SIAM J. Math. Anal. 25 (1994), no. 5, 1259-1268. doi: 10.1137/S0036141093249080 – reference: Lenzmann, E. Uniqueness of ground states for pseudorelativistic Hartree equations. Anal. PDE 2 (2009), no. 1, 1-27. doi: 10.2140/apde.2009.2.1 – reference: Carmona, R.; Masters, W. C.; Simon, B. Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91 (1990), no. 1, 117-142. doi: 10.1016/0022-1236(90)90049-Q – reference: McLeod, K. Uniqueness of positive radial solutions of Δ u + f(u) = 0 in Rn. II. Trans. Amer. Math. Soc. 339 (1993), no. 2, 495-505. doi: 10.2307/2154282 – reference: Abdelouhab, L.; Bona, J. L.; Felland, M.; Saut, J.-C. Nonlocal models for nonlinear, dispersive waves. Phys. D 40 (1989), no. 3, 360-392. doi: 10.1016/0167-2789(89)90050-X – reference: Albert, J. P.; Bona, J. L.; Saut, J.-C. Model equations for waves in stratified fluids. Proc. Roy. Soc. London Ser. A 453 (1997), no. 1961, 1233-1260. doi: 10.1098/rspa.1997.0068 – reference: Silvestre, L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60 (2007), no. 1, 67-112. doi: 10.1002/cpa.20153 – reference: Weinstein, M. I. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), no. 3, 472-491. doi: 10.1137/0516034 – reference: Bañuelos, R.; Kulczycki, T. The Cauchy process and the Steklov problem. J. Funct. Anal. 211 (2004), no. 2, 355-423. doi: 10.1016/j.jfa.2004.02.005 – reference: Ma, L.; Zhao, L. Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. 195 (2010), no. 2, 455-467. doi: 10.1007/s00205-008-0208-3 – reference: Coffman, C. V. Uniqueness of the ground state solution for Δ u-u + u3 = 0 and a variational characterization of other solutions. Arch. Rational Mech. Anal. 46 (1972), 81-95. doi: 10.1007/BF00250684 – reference: Kulczycki, T.; Kwaśnicki, M.; Mał;ecki, J.; Stos, A. Spectral properties of the Cauchy process on half-line and interval. Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 589-622. doi: 10.1112/plms/pdq010 – reference: Fabes, E. B.; Kenig, C. E.; Serapioni, R. P. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7 (1982), no. 1, 77-116. doi: 10.1080/03605308208820218 – reference: Krieger, J.; Lenzmann, E.; Raphaël, P. Nondispersive solutions to the L2-critical half-wave equation. Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61-129. doi: 10.1007/s00205-013-0620-1 – reference: Chang, S.-Y. A.; Gonzàlez, M. d. M. Fractional Laplacian in conformal geometry. Adv. Math. 226 (2011), no. 2, 1410-1432. doi: 10.1016/j.aim.2010.07.016 – reference: Chen, W.; Li, C.; Ou, B. Qualitative properties of solutions for an integral equation. Discrete Contin. Dyn. Syst. 12 (2005), no. 2, 347-354. – reference: Weinstein, M. I. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1982/83), no. 4, 567-576. – reference: Kwong, M. K. Uniqueness of positive solutions of Δ u-u + up = 0 in Rn. Arch. Rational Mech. Anal. 105 (1989), no. 3, 243-266. doi: 10.1007/BF00251502 – reference: Reed, M.; Simon, B. Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978. – reference: Amick, C. J.; Toland, J. F. Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane. Acta Math. 167 (1991), no. 1-2, 107-126. doi: 10.1007/BF02392447 – reference: Li, Y. Y. Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. (JEMS) 6 (2004), no. 2, 153-180. – reference: Graham, C. R.; Zworski, M. Scattering matrix in conformal geometry. Invent. Math. 152 (2003), no. 1, 89-118. doi: 10.1007/s00222-002-0268-1 – reference: Kwaśnicki, M. Eigenvalues of the fractional Laplace operator in the interval. J. Funct. Anal. 262 (2012), no. 5, 2379-2402. doi: 10.1016/j.jfa.2011.12.004 – reference: Chen, W.; Li, C.; Ou, B. Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59 (2006), no. 3, 330-343. doi: 10.1002/cpa.20116 – reference: Merle, F.; Raphael, P. The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. (2) 161 (2005), no. 1, 157-222. doi: 10.4007/annals.2005.161.157 – volume: 7 start-page: 77 issue: 1 year: 1982 end-page: 116 article-title: The local regularity of solutions of degenerate elliptic equations publication-title: Comm. Partial Differential Equations – volume: 105 start-page: 243 issue: 3 year: 1989 end-page: 266 article-title: Uniqueness of positive solutions of Δ ‐ + = 0 in publication-title: Arch. Rational Mech. Anal. – volume: 40 start-page: 360 issue: 3 year: 1989 end-page: 392 article-title: Nonlocal models for nonlinear, dispersive waves publication-title: Phys. D – volume: 167 start-page: 107 issue: 1‐2 year: 1991 end-page: 126 article-title: Uniqueness and related analytic properties for the Benjamin‐Ono equation—a nonlinear Neumann problem in the plane publication-title: Acta Math. – volume: 87 start-page: 567 issue: 4 year: 1982/83 end-page: 576 article-title: Nonlinear Schrödinger equations and sharp interpolation estimates publication-title: Comm. Math. Phys. – volume: 211 start-page: 355 issue: 2 year: 2004 end-page: 423 article-title: The Cauchy process and the Steklov problem publication-title: J. Funct. Anal. – volume: 329 start-page: 383 issue: 1 year: 2014 end-page: 404 article-title: Uniqueness and nondegeneracy of positive solutions of (−Δ) + = in ℝ when is close to 1 publication-title: Comm. Math. 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| Snippet | We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ)s with s ∊ (0,1) for any... We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (−Δ) s with s ∊ (0,1) for any... We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian (...)^sup s^ with s... (0,1) for... |
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| SubjectTerms | Laplace transforms Linear equations Nonlinear equations Schrodinger equation |
| Title | Uniqueness of Radial Solutions for the Fractional Laplacian |
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